ROTATION AND DYNAMICS FOR SIMPLE SOLENOIDAL MAPS OF TORI
by
Mark Tyler Mathison
A dissertation submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Mathematics
MONTANA STATE UNIVERSITY
Bozeman, Montana
April 2012
c
COPYRIGHT
by
Mark Tyler Mathison
2012
All Rights Reserved
ii
APPROVAL
of a dissertation submitted by
Mark Tyler Mathison
This dissertation has been read by each member of the dissertation committee and
has been found to be satisfactory regarding content, English usage, format, citations,
bibliographic style, and consistency, and is ready for submission to The Graduate
School.
Dr. Jaroslaw Kwapisz
Approved for the Department of Mathematics
Dr. Kenneth L. Bowers
Approved for The Graduate School
Dr. Carl A. Fox
iii
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Mark Tyler Mathison
April 2012
iv
TABLE OF CONTENTS
1. INTRODUCTION.......................................................................................1
History .......................................................................................................1
Quasi-Periodically Forced Circle Maps ....................................................3
Overview ....................................................................................................6
2. SIMPLE SOLENOIDAL MAPS AND THEIR ROTATION SETS ................ 12
SSM Definition and General Facts .............................................................. 12
SSM with Fixed Point and Non-Negative Displacement ............................... 24
Flowable SSM with a Stopped Point........................................................... 29
Existence and Uniqueness for SSM ODE ................................................... 42
Non-flowable SSM with a Stopped Point ..................................................... 47
Approximating the Map with Flows...................................................... 51
Regularity Near p0 , and Integrals.......................................................... 56
Conclusion of Proof of Theorem 7......................................................... 61
3. INVARIANT MEASURES......................................................................... 64
Coexistence and Cohomological Equation ................................................... 65
Proof of Theorem 8 ................................................................................... 69
Existence Examples ............................................................................. 81
Minimality (Almost) ............................................................................ 90
4. FIXED POINT FREE MAPS .................................................................... 96
Bounded Deviation.................................................................................... 96
Flow Equivalence ...................................................................................... 99
Suspension Flow .................................................................................100
Establishing T ....................................................................................103
ρ-Bounded T ............................................................................................108
Semi-Conjugacy .......................................................................................116
APPENDICES ..............................................................................................123
APPENDIX A: CONTINUED FRACTIONS .............................................124
APPENDIX B: OSTROWSKI’S β-NUMERATION....................................131
REFERENCES CITED..................................................................................142
v
LIST OF FIGURES
Figure
Page
1. Orbit of time-one-map of special flow under r(x) .....................................9
2. Invariant set with non-zero and non-full lebesgue measure...................... 10
3. Cartesian graph of g and corresponding interval exchange on T .............. 49
4. A map that is not the time one map of a flow ....................................... 50
5. Elongation of U0 .................................................................................. 92
6. Fact 11 ..............................................................................................100
7. Construction of G̃t ..............................................................................102
8. Flow lines do not cross........................................................................104
9. Construction of T̃ ...............................................................................106
10. Bounding |˜l − ˜ln | ................................................................................111
11. Mutual bounding on the deviation of F and T .....................................112
12. Ĥ is Equivariant under addition of (0, 0, 1)...........................................121
13. Rβ .....................................................................................................127
14. Finding continued fraction representation.............................................129
15. Rotations bounded away from zero ......................................................140
vi
ABSTRACT
The rotation number for a circle map has provided a complete and useful classification for that class of maps. In higher dimensions there is still progress to be
made towards obtaining a more complete understanding of the relationship between
the map and its average rotation.
In this dissertation, we explore a class of homeomorphisms on the d-dimensional
torus Td that preserve each leaf of a foliation of the torus into parallel lines densely
winding on Td . First the rotation sets of such maps are explored, with particular
emphasis on those maps that have a single fixed point; zero is necessarily an element
of those rotation sets. Conditions are found that show when these maps have a
non-trivial rotation set.
When such maps, with non trivial rotation sets, are created as the time-one-map
of a flow it is shown that the existence of merely two, or infinitely many ergodic
measures is connected to the solvability of a cohomological equation. An example, of
the infinitely many ergodic measure case, is provided.
Finally, we explore on T2 maps without a fixed point that happen to also have
a point whose orbit has bounded deviation from the mean rotation. Such maps are
seen to be akin to circle maps with irrational rotation number; the irrationally sloped
foliation leads to the map being semi-conjugate to an irrational translation of T2 .
1
CHAPTER 1
INTRODUCTION
History
A naturally arising question when studying a map f : X → X is if there is a
“typical behavior.” In particular, if f can be lifted to f˜ : X̃ → X̃ where X̃ is a linear
space covering X we can ask about average velocities under f˜. Precisely, we want to
study when this limit converges
f˜i (x̃) − x̃
.
i→∞
i
ρ(f˜, x̃) := lim
Studying orientation-preserving homeomorphisms f : T → T, where T := R/Z is
the circle obtained as the quotient of the reals by its subgroup of integers, it is natural
to consider a projection π : R → T = R/Z and lifts f˜ : R → R where f ◦ π = π ◦ f˜. In
this context Poincaré [1] discovered that ρ(f˜, x̃) exists for all x̃ ∈ R, is independent
of x̃ and the rotation number of f ,
ρ(f ) :=π(ρ(f˜, x̃)),
is independent of our choice of lift f˜.
The rotation number of a circle homeomorphism gives a very complete picture of
the behavior of f . Poincaré knew that when ρ(f ) is rational, then f has a periodic
point, and all periodic orbits have the same period. He also knew that when ρ(f ) is
2
irrational, then f is semi-conjugate to the rotation Rρ(f ) via a continuous surjection
h. When f is a diffeomorphism and f ′ is of bounded variation, the Denjoy Theorem
[2, 3] furthers Poincaré’s result, h is a homeomorphism and f is conjugate to the
irrational rotation Rρ(f ) .
The existence of a rotation number for circle maps has also been shown under
weaker conditions. In [4] continuous non-decreasing degree one maps f are shown to
have a rotation number. In [5] we learn that continuity is not required for a unique
rotation number, all that is required is to have a degree one f that is non-decreasing.
Non-decreasing seems to be the key requirement as a continuous degree one map f
that does not satisfy the non-decreasing requirement is shown in [6] to have, instead
of a rotation number, a rotation set (in this case the set is an interval) which is defined
as the closure of
(
f˜i (x̃) − x̃
lim sup
i
i→∞
)
.
x̃∈R
Later [7] showed that this set was (already) closed.
Only in dimension one do general orientation-preserving homeomorphisms f :
Td → Td produce an invariant (over choice of lift and x̃) vector ρ(f˜, x̃) without
imposing additional constraints. So, as with the non-monotonic lifts of circle maps,
we extend our study from a single rotation vector to a rotation set. One “natural”
way to define a rotation set is referred to as the pointwise rotation set, ρp (f˜). It can
be constructed by gathering together all convergent subsequences of the set {(f˜i (x̃) −
x̃)/i}∞
i=1 for all x̃. Another approach was introduced by Misiurewicz and Ziemian in
3
[8]. There they define the rotation set ρ(f˜) for a continuous map f˜ : Rd → Rd as the
set of limits,
f˜ni (xi ) − xi
i→∞
ni
ν := lim
taken over all pairs of sequences {xi } ∈ Rd and {ni } ∈ N, with limi→∞ ni = ∞, for
which the limit exists. This rotation set, ρ(f˜), will be the primary focus of Chapter 2
of this paper, in part due to the following theorem which describes two of the nice
properties that this rotation set has.
Theorem 1 For continuous f : Td → Td homotopic to the identity ρ(f˜) is compact
and connected.
One should note that for continuous f : Td → Td homotopic to the identity with
d 6= 1, ρp (f˜) need not be compact [9] or connected [10].
We briefly mention one more rotation set, ρmes (f˜). Elements of this set are given
by each
Z
f˜(y) − y dµ
where y ∈ π −1 (x) for x ∈ Td and µ is a f -invariant ergodic probability measure of
Td . All three of these rotation sets share some elements [8] in particular ρmes (f˜) ⊂
ρp (f˜) ⊂ ρ(f˜).
Quasi-Periodically Forced Circle Maps
Trying to classify lifts of homeomorphic maps homotopic to the identity on Td
(for d > 1) in general does not work out as nicely as in the one-dimensional case.
4
As previously mentioned, the limits limi→∞ (f˜i (x̃) − x̃)/i are not the same for all x̃
in general, worse yet these limits need not even exist. So instead of general results
much work is done using homeomorphisms with additional constraints. Some nice
results (like the classification of the transitive sets, by the number of minimal sets
they contain, given in [11]) can be obtained by restricting to certain skew products.
In particular, when Tθ : T → T is an orientation preserving homeomorphism for every
θ ∈ T and θ 7→ θ + ω is minimal, we can create a quasi-periodically forced circle map
on T2 = T × T given by
T (θ, x) = (θ + ω, Tθ (x)).
(1.1)
These types of maps, denoted by T ∈ Thom , have a fibrewise rotation number,
T̃θi (x̃) − x̃
,
i→∞
i
ρT̃ = lim
where T̃θi (x̃) = πx T i (θ, x). Furthermore (from [12]), ρT :=ρT̃ mod 1 is invariant under
choice of lift, choice of (starting) fibre map T̃θi : R → R, and starting point x̃ ∈ R. So
the lift T̃ (θ̃, x̃) = (θ̃ + ω, T̃θ (x̃)) has a well defined rotation vector
T̃ i (θ̃, x̃) − (θ̃, x̃)
.
i→∞
i
(ω, ρT ) = lim
A key boundedness property that occurs in the one-dimensional case leads to a
uniform bound on deviations from uniform rotation. In particular when f˜ is a lift of
a homeomorphism on the circle with rotation number ρ we have
|f˜n (x) − x − nρ| ≤ 1 for all n ∈ N, x ∈ R.
5
A similar bound, on deviations from uniform rotation, is not present for general quasiperiodically forced circle maps. In [13] the authors call the orbits of those maps that
do meet such a property ρ-bounded.
Definition 1 If T̃ is a lift of T ∈ Thom we say the orbit of (θ̃, x̃) is ρ-bounded if there
exists a constant C > 0 such that
|T̃θn (x̃) − x̃ − nρT̃ | ≤ C
∀n ∈ N,
(1.2)
This property is independent of choice of lift.
In [14], slightly extended in [13], it is shown that
Theorem 2 If there exists one ρ-bounded orbit, then all orbits are ρ-bounded and the
constant C in (1.2) can be chosen uniformly for all (θ, x) ∈ T2
Because of this theorem it is well-defined to refer to a map T ∈ Thom as being ρbounded if and only if any of its orbits are ρ-bounded. These ρ-bounded maps have
a classification very similar to the classification of circle maps. In particular we have
Theorem 3 [adapted from Theorem 3.1 in [13]] Suppose T ∈ Thom is ρ-bounded.
If ρT and ω are rationally independent then T is semi-conjugate to the irrational
torus-translation Rω,ρT . Furthermore, the semi-conjugacy h can be chosen so that it
is fibre-respecting (that is, π1 ◦ h = π1 ) and all fibre maps hθ are order-preserving
circle maps.
6
The second condition is of particular interest to us and will be a key factor in the
results for Chapter 4.
Overview
This paper deals with a class of “dimension 1.5” toral maps which are given by
Definition 2 A continuous map F : Td → Td is a Simple Solenoidal Map (SSM) if
and only if there exists a lift of F , F̃ : Rd → Rd , and a vector α = (α1 , . . . , αd ) ∈ Rd
whose components are independent over Q so that, for any p̃ ∈ Rd ,
F̃ ({p̃ + t · α}t∈R ) = {p̃ + t · α}t∈R .
The lines {p̃+t·α} are lifts to Rd of families of orbits under F . Each family, consisting
of many orbits, is dense on Td . This denseness together with continuity of the SSM
combine to allow us to think of each of the families, even those that lift to a different
line in Rd , as interconnected. Thus by restricting to a single family of orbits and
studying a particular one-dimensional lift, fp : R → R, we may gain insight into the
entire d-dimensional SSM.
Chapter 2 of this thesis focuses on determining whether the rotation set for a
SSM contains only a single element or a larger set. Our study is quickly divided into
two classes, those SSM that have a fixed point and those that do not. In the later
case, with a few moderate assumptions on fp , F̃ has a unique (non-zero) average
displacement vector.
In contrast, the existence of a single fixed point in just one family of orbits assures
7
that zero is an element of the rotation set and propagates as recurring “slowed-down”
regions in each of the families of orbits. To find a rotation set with more than a single
element the slow-down needs to be controlled to ensure that some orbits maintain
a positive average displacement. For SSMs constructed as time-one-maps of flows
the criterion for this has already been established and is part of the mathematical
“folklore” in the field. We consolidate some of those results and supply proofs missing
from the literature in Theorem 5.
For those SSM that do not necessarily emerge as time-one-maps of flows we establish the following
Main Result 1 Suppose that F is an SSM (with single stopped point p0 ) given by
F (p) := p + Φ · α where Φ : Td → [0, ∞) and Φ−1 (0) = {p0 }. Assume that Φ is C 1
and convex on a neighborhood of p0 . Then the following are equivalent:
(i) For all p ∈ Td , ρ(fp ) is not a point.
fpn (x) − x
exists and is not 0.
n→∞
n
(ii) There is p ∈ Td such that for some x ∈ R, lim
fpn (x) − x
exists and is not 0.
(iii) There is p ∈ T such that for all x ∈ R, lim
n→∞
n
Z
1
(iv)
|dr̃| < +∞ (integrated according to the ordinary Lebesgue measure).
[0,1]d Φ̃(r̃)
d
Chapter 3 looks at invariant measures for Simple Solenoidal Maps (F ) that arise
as the time-one-maps of flows (F t ) that have a rotation set with more than one
element. To address this we first remove from Td the set L of points that lift to the line
containing the fixed point. On a codimension one subtorus K, where K ×{0} ⊂ Td \L,
8
there is a first return map given by x 7→ x + β. We define the return time function
r : K → [0, ∞) by r(x) = t where t > 0 is the first return time to K × {0}. The
number of ergodic measures for the SSM F is then seen to be connected to the
solvability of a certain cohomological equation.
Theorem 8 The time-one-map F = F 1 has either one or uncountably many ergodic
measures on Td \L. The latter case takes place if and only if there is a non-zero
rational constant κ and a measurable ĥ : Td−1 → S1 such that, for Lebesgue a.e.
x ∈ Td−1 ,
ĥ(x + β) = e2πiκr(x) · ĥ(x).
Since the existence of SSM’s with return times that are cohomologous to a
rational constant are by no means obvious we supply
Main Result 2 There exists a flowable SSM F on T2 with a single stopped point,
p0 , for which there are uncountably many distinct ergodic invariant measures µ. For
such SSM the only closed invariant sets are {p0 } and Td . In particular, any bi-infinite
F -orbit in Td \ L is dense in all of Td .
The proof of existence entails an explicit construction of a pathological example
(in dimension 2) for which the associated cohomological equation is solvable. The
SSM generated therein is one that has multiple ergodic measures. To give an idea of
what some simple invariant sets look like under this map, Figure 1 shows an orbit
of the time-one-map of the special flow representation of such an SSM. The special
flow acts horizontally with base space depicted as the segment [0, 1] on the vertical
9
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Figure 1. 15,000 (on left) and 60,000 (on right) points in the orbit of (.4, .4) under
the iteration
of the time-one-map of the special flow with base rotation (vertical axis)
√
(1 + 5)/2 under r(x) constructed via R(z) from (3.12) (with b=3, p=.75) and then
normalized to have an integral of 1. r(x) is cohomologous to the constant 1 via a
measurable map h(x). The jagged nature of the image is a result of the densely
distributed discontinuities of h .
axis with endpoints identified. The ceiling (bounding graph on the right) function
r(x) has an asymptote at 0 (corresponding to the fixed point of the SSM) and will
be constructed as part of the existence proof.
When we “thicken” the F -orbit by
taking the orbit of a line segment parallel to the flow we, can observe in Figure 2
what appear to be invariant sets with non-zero (nor full) lebesgue measure.
Indeed Corollary 9 shows that such SSM are not ergodic with respect to the
Lebesgue measure; this condition becomes interesting when taken in conjunction with
the second half of Main Result 2 implying that the invariant sets, whose approximate
representation is given by the jagged graphs in Figure 1, are in fact dense in the whole
phase space. We note that our base rotation uses β that is an irrational of constant
type, which is not well approximated by rationals unlike similar examples due to
10
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Figure 2. A repeat of the 15k iterates in the time-one-map orbit of (.4, .4) under
r(x) shown in figure 1 followed by a ”thickening” to a .1 measure, .5 measure and .9
measure .
Furstenberg [15] and others [16, 17, 18] who use well approximable rotations.
Chapter 4 looks at fixed point free Simple Solenoidal Maps of T2 that have
bounded deviation from the mean rotation. That is, for any p̃ there exists C > 0
such that for any n ∈ Z
kF̃ n (p̃) − p̃ − nρk < C.
SSMs with this property are analogous to the ρ-bounded maps meeting condition 2
of Theorem 3. Indeed we show
11
Main Result 3 If an SSM, F : T2 → T2 , is a fixed point free homeomorphism with
bounded deviation, then F is semi-conjugate to an irrational translation.
In an independent recent development [19], Jäger and Aliste-Prieto proved this result
for a more general class of solenoidal systems. Our proof uses a flow equivalence
between the SSM F and T , a ρ-bounded quasi-periodically forced map of a torus.
The majority of the work here is in establishing
Lemma 15 F has bounded deviation if and only if T has bounded deviation. Furthermore the bounding constant C may be chosen to be independent of the choice of
p̃ = (θ̃, x̃).
The Lemma provides the framework by which we study the relationship between the
bounds (from mean rotation) of the two flow equivalent maps. Once that is done
we merely “chase” the semi-conjugacy established by Theorem 3 back through the
flow equivalency to the space our map inhabits. The argument in [19] proceeds along
different more direct lines.
12
CHAPTER 2
SIMPLE SOLENOIDAL MAPS AND THEIR ROTATION SETS
SSM Definition and General Facts
Let Td = Rd /Zd be the d-dimensional torus obtained as the quotient of the ddimensional Cartesian space by the integral lattice. Denote by π : Rd → Td the
canonical quotient map. We are interested in homeomorphisms that preserve each
leaf of a foliation of the torus into parallel lines densely winding on Td . Such homeomorphisms belong to a somewhat broader class of Simple Solenoidal Maps (SSM),
defined as follows.
Definition 2 A continuous map F : Td → Td is a Simple Solenoidal Map (SSM)
if there exists a lift of F , F̃ : Rd → Rd , and a vector α = (α1 , . . . , αd ) ∈ Rd whose
components are independent over Q so that, for any p̃ ∈ Rd ,
F̃ ({p̃ + t · α}t∈R ) = {p̃ + t · α}t∈R .
The independence condition on (α1 , . . . , αd ) means that α is non-resonant in the
sense that there cannot be ki ∈ Z that are not all zero and k1 α1 + . . . + kd αd = 0.
This is exactly when the canonical projection π sends any line in the direction of α,
p̃ + Rα, to a copy of R densely immersed in Td . Such projections are the leaves of
a foliation of Td . Note that F is surjective since the image of any single leaf, being
13
the leaf itself, is dense in Td . The definition of SSMs does not insist on injectivity of
F but it will be required later for our key results. It is easy to see that SSM with
the same α form a semigroup under the composition operation. In particular, the
iterates F n are SSM for n ∈ N. However, we should point out the following fact that
is a basis of a good rotation set theory.
Lemma 1 Any SSM F is homotopic to the identity. In particular, for any lift F̃ , we
have
F̃ (p̃ + v) = F̃ (p̃) + v,
p̃ ∈ Rd , v ∈ Zd .
(2.1)
Proof of Lemma 1:
Since F is a continuous map of Td , there is a d × d matrix A with integer entries
such that F̃ (p̃ + v) = F̃ (p̃) + Av for any p̃ ∈ Rd , v ∈ Zd and F̃ a lift of F . (This A is
the matrix of the action of F on the first homology of Td with the basis given by the
loops in the directions of the circle factors in Td ≃ T1 × . . . × T1 .) F is homotopic to
the identity if and only if A is the identity matrix, A = I, in which case the homotopy
is given by F̃t (p̃) := p̃ + t(F̃ (p̃) − p̃).
Since F̃ is an SSM, we see that when v ∈ Zd both
F̃ (p̃ + v + tα) ∈ p̃ + v + Rα
and
F̃ (p̃ + tα) + Av ∈ p̃ + Av + Rα
14
so that
Av − v ∈ Rα.
This means that A−I is a matrix with integer entries and the column space contained
in Rα. Due to the non-resonance condition, this implies A − I = 0.
2
By restricting an SSM F to individual leaves, one obtains 1-dimensional maps.
To the extent that F can be recovered from any one of these maps (by the process of
continuous extension), F is essentially 1-dimensional. Let us introduce some notation
to formalize this simple but crucial observation.
For p̃, q̃ ∈ Rd we define an equivalence relation ∼ of being on the same line in the
direction α by
p̃ ∼ q̃
if and only if
q̃ = p̃ + aα
for some a ∈ R.
We denote by f˜p̃ : [p̃] → [p̃] the restriction of F̃ to the line constituting the equivalence
class [p̃]. Then, letting σp̃ be the immersion
σp̃ : R ∋ t 7→ p̃ + tα ∈ [p̃],
we define fp̃ : R → R by
fp̃ :=σp̃−1 ◦ f˜p̃ ◦ σp̃ .
(2.2)
In view of (2.1), we have fp̃ = fp̃+v for v ∈ Zd so, given p ∈ Td , we can also
unambiguously define fp := fp̃ where p̃ ∈ Rd is a lift of p.
15
As anticipated, the vector α and just one of the maps fp̃ from (2.2) uniquely
determine F and F̃ , as follows. First, fp̃ determines f˜p̃ = σp̃ ◦ fp̃ ◦ σp̃−1 on [p̃] and
thus also the restriction of F to the leaf π([p̃]). Because π([p̃]) is dense in Td and F
is continuous, F is uniquely determined on all of Td . Finally, F̃ is recovered as the
unique lift of F that sends p̃ to f˜p̃ (p̃).
In light of this fact, for any p, q ∈ Td , the function fp is somehow encoded in fq .
In fact, we shall prove below that fp is a (uniform) limit of translates fq (· + tn ) − tn
for a suitable sequence (tn ) ⊂ R. (Later we look at a situation where fp̃ has a fixed
point but (for q̃ 6= p̃) fq̃ does not, so it need not be that fp̃ and fq̃ are translates of
each other.) To begin explaining this, one starts with a simple observation:
Fact 1 For all p ∈ T, x, t ∈ R we have fp (x + t) − t = fp+tα (x).
Proof of Fact 1:
By using that p̃ + tα ∼ p̃ and thus f˜p̃+tα = f˜p̃ , we have
σp̃ (fp+tα (x) + t) = p̃ + tα + (fp+tα (x))α = σp̃+tα (fp+tα (x))
= f˜p̃+tα (σp̃+tα (x)) = f˜p̃ (p̃ + tα + xα)
= f˜p̃ (σp̃ (x + t)).
Thus
fp+tα (x) + t = σp̃−1 (f˜p̃ (σp̃ (x + t)) = fp (x + t),
(2.3)
and fp (x + t) − t = fp+tα (x) as desired.
2
16
Let X be the space of continuous functions, f : R → R such that supx∈R |f (x) −
x| < ∞, with distance given by dist(f, g) := supx∈R |f (x) − g(x)|.
Lemma 2 The map Td → X given by p 7→ fp is continuous.
Proof of Lemma 2:
Take p ∈ Td . Fix a small ǫ > 0. By uniform continuity of F̃ , there is δ > 0
(for technical reasons we also require δ < ǫ|α|/2) such that, for p̃, q̃ ∈ Rd , |p̃ − q̃| < δ
implies |F̃ (p̃) − F̃ (q̃)| < ǫ|α|/2.
Now, let p̃ ∈ Rd be a lift of p. Suppose that q ∈ Td is in the δ-neighborhood1 of
p. Our goal is to show that |fp (x) − fq (x)| < ǫ.
There is a lift q̃ ∈ Rd of q such that |q̃ − p̃| < δ. In this way |q̃ + xα − (p̃ + xα)| < δ
and thus |F̃ (p̃ + xα) − F̃ (q̃ + xα)| < ǫ|α|/2 for all x ∈ R. To see that this implies
|fp̃ (x) − fq̃ (x)| < ǫ, we estimate
fp̃ (x) − fq̃ (x) |α| − δ ≤ fp̃ (x)α − fq̃ (x)α − |q̃ − p̃|
≤ (fp̃ (x)α − fq̃ (x)α) − (q̃ − p̃)
= p̃ + fp̃ (x)α − (q̃ + fq̃ (x)α) = σp̃ (fp̃ (x)) − σq̃ (fq̃ (x))
= f˜p̃ (σp̃ (x)) − f˜q̃ (σq̃ (x)) = f˜p̃ (p̃ + xα) − f˜q̃ (q̃ + xα)
= F̃ (p̃ + xα) − F̃ (q̃ + xα) < ǫ|α|/2.
Thus, for x ∈ R, |fp̃ (x) − fq̃ (x)| < (ǫ|α|/2 + δ)/|α| < ǫ since δ < ǫ|α|/2.
2
1
We use the Riemannian metric on Td induced from the Euclidean metric on Rd .
17
Corollary 1 Given p, q ∈ Td , there is (tn ) ⊂ R such that fp (· + tn ) − tn converges
uniformly to fq .
Proof of Lemma 1:
By the density of the leaves, there is (tn ) ⊂ R such that p + tn α → q (where the
addition is performed modulo Zd ). It remains to observe that fp+tn α = fp (· + tn ) − tn
(by Fact 1) and invoke Lemma 2 to get uniform convergence fp (· + tn ) − tn → fq .
2
Corollary 2 For any p, q ∈ Td , ρ(fp ) = ρ(fq ).
As we shall see from the proof below, at the heart of the corollary is the upper
semi-continuity of the rotation set as a function of the map, which is an easy consequence of the particular definition of the rotation set we adopted. It has been noted
already in [8], Theorem 2.10.
Proof of Corollary 2:
Take ν ∈ ρ(fp ). There are sequences (xi ) ∈ R and (ni ) ∈ N with ni → ∞ so
that
fpni (xi ) − xi
= ν.
i→∞
ni
lim
For each i ∈ N, by applying Corollary 1 to ni -th iterates, we can find ti ∈ R such
that
sup |fqni (x + ti ) − ti − fpni (x)| < 1.
x
18
In particular,
Hence
ni
fq (xi + ti ) − (xi + ti ) fpni (xi ) − xi ≤ 1.
−
ni
ni
ni
fqni (xi + ti ) − (xi + ti )
lim
=ν
i→∞
ni
so ν ∈ ρ(fq ). This proves ρ(fq ) ⊂ ρ(fp ). The opposite inclusion ρ(fp ) ⊂ ρ(fq ) holds
for the same reason.
2
In a similar vein are the following two corollaries, the proofs of which we skip.
Corollary 3 For any p, q ∈ Td , if fp is non-decreasing then so is fq .
Corollary 4 If there exists p1 ∈ Td for which fp1 (x) − x ≥ 0 for all
2
x ∈ R, then
for any p ∈ Td we have fp (x) − x ≥ 0 for all x ∈ R.
The quantity fp (x) −x is called the displacement of x and the last corollary shows
that the hypothesis of having non-negative displacement is independent of p. Similar
is the case for non-positive displacement but we will not bother with considering this
as separate case because the conjugation of F by p 7→ −p reduces it to the nonnegative case. We are also not that interested in the case when the displacement does
not have a definite sign as then the dynamics of fp are rather boring.
To see why, we make use of the fact that for any p the displacement function
x 7→ fp (x) − x is almost periodic in the sense of Bochner and Bohr [20]. That
2
in fact, all x > x0 for some x0 ∈ R is sufficient
19
is, for any sequence (tn ) ∈ R a subsequence of (fp (x + tn ) − (x + tn )) converges
uniformly3 . To see this, observe that Corollary 1 gives us the uniform convergence
(fp (x + tn ) − (x + tn )) → fq (x) − x when we let q be any limit point of (p + tn α) ∈ T.
This allows us to show the following:
Proposition 1 If F is a SSM and, for some p ∈ Td , fp (x) − x changes sign, then
it does so for all p ∈ Td . Moreover, in such cases, if fp is non-decreasing then,
ρ(fp ) = {0} and, for any p ∈ Td , there is a (bi-infinite) sequence (xk ) ⊂ R such that
fp (xk ) = xk with limk→−∞ xk = −∞ and limk→+∞ xk = +∞.
Proof of Proposition 1:
Since fp (x) − x changes sign we may find ε > 0 and − x0 , + x0 ∈ R where
fp (− x0 ) − − x0 < −ε < ε < fp (+ x0 ) − + x0 .
For any q ∈ Td , by Corollary 1, there is a sequence (tn ) ∈ R with fq (· + tn ) − tn
converging uniformly to fp (·) so that
fq (− x0 + tn ) − (− x0 + tn ) → fp (− x0 ) − − x0 < −ε
and
fq (+ x0 + tn ) − (+ x0 + tn ) → fp (+ x0 ) − + x0 > ε.
As such (by the uniform continuity) we can find M > 0 so that when |n| > M we
have
3
This is the Bochner definition which [20] shows is equivalent to the Bohr definition we will use
momentarily.
20
fq (− x0 + tn ) − − x0 + tn < 0
and
fq (+ x0 + tn ) − + x0 + tn > 0,
so fq (x) − x changes signs. Since q was arbitrary the displacement function fq (x) − x
changes signs for all q ∈ Td .
To show ρ(fp ) = 0 we define two sequences of points, one with positive displacement, the other with negative displacement. Since the displacement function is
almost periodic we may find4 a length L(ε) such that any interval of the real line with
length L(ε) contains at least one point whose displacement is in the ε-neighborhood
of fp (− x0 )− − x0 and another point (in L(ε)) with displacement in the ε-neighborhood
of fp (+ x0 ) − + x0 . That is, any interval with length at least L(ε) contains both points
with positive displacement and points with negative displacement. We create such an
interval L0 with − x0 , + x0 ∈ L0 and length |L0 | ≥ |L(ε)|. The collection of intervals
{Lk :=L0 + k|L0 | : k ∈ Z}
monotonically tiles the real line. From each tile we may select two points − xk , + xk ∈
Lk where
fp (− xk ) − − xk < 0
and
fp (+ xk ) − + xk > 0.
For any ν ∈ ρ(fp ) we take sequences (ni ) and (xi ) (with limi→∞ ni = ∞) where
fpni (xi ) − xi
= ν.
i→∞
ni
lim
If any xi happens to have displacement zero then it is a fixed point of fp . For xi
that are not fixed points the displacement is nonzero, which enables us to find a fixed
4
This is the Bohr definition of almost periodic.
21
point on either side of xi . These fixed points may be chosen a bounded distance
apart (from each other) enabling us to force the above limit to zero. We carry out
the details below.
For any xi with nonzero displacement we find k so that either − xk < xi < − xk+1
(when the displacement of xi is positive) or + xk < xi < + xk+1 (when the displacement
of xi is negative). In either case continuity of x → fp (x) allows us to find x−
i ∈
(−/+ xk , xi ) and x+
i ∈ (xi , −/+ xk+1 ) with
−
fp (x−
i ) − xi = 0
and
+
fp (x+
i ) − xi = 0.
+
(That is x−
i and xi are the promised fixed points (of fp ) on either side of xi .) Since
fp is non-decreasing we have
+
fpni (x−
fpni (xi ) − xi
fpni (x+
x−
x+ − x−
i ) − xi
i ) − xi
i − xi
i
<
≤
≤
< i
.
ni
ni
ni
ni
ni
+
As x−
i , xi ∈ Lk
S
Lk+1 we have
ni
fp (xi ) − xi 2|L0 |
≤
.
ni
ni
(2.4)
Since we trivially also have (2.4) for those xi where the displacement is zero we find
that
ni
fp (xi ) − xi ≤ lim 2|L0 | = 0,
|ν| = lim i→∞ ni
i→∞
ni
and we have ρ(fp ) = 0 as desired.
To finish we note that, similar to finding the fixed points above, in each Lk ,
between − xk and + xk there is a point xk with fp (xk ) − xk = 0. Since the Lk ’s tile the
22
line monotonically, limk→−∞ xk = −∞ and limk→+∞ xk = +∞ as desired.
2
Having now seen that the restriction to fp with non-negative displacement is
desirable (for the possibility of a nonzero rotation set), the natural question to ask
is if it is also beneficial to require that the (non-negative) displacement be strictly
positive. This turns out to be a misleading condition since Proposition 2 will show
that unless the displacements are bounded away from zero, F (and thus at least one
fp ) will have a fixed point.
Proposition 2 For a SSM F , inf p∈Td fp (0) = 0 if and only if there exists p0 ∈ Td
with F (p0 ) = p0 and p ∈ Td with fp (x) − x ≥ 0 for all x ∈ R.
We delay the proof (until the end of this section) and note that imposing a
(strictly) positive displacement for all fp , gives the seemingly stronger condition that
the displacement of fp is bounded away from zero. If we (also) require that fp be
non-decreasing (as we often will) then the resulting rotation set will be a one-pointset. So in the general exploration of (interesting) rotation sets that follows, we will
not impose this (positive displacement) condition.5
Theorem 4 For a SSM F without fixed points, if fp is non-decreasing and has nonnegative displacement then F̃ has a unique (nonzero) average displacement vector
ρ = ρ(fp ) · α.
5
However later, in Chapter 4 to establish a single rotation number (given as the lone element in
its rotation set), we will bound our displacement away from zero.
23
Proof of Theorem 4:
For any ρ ∈ ρ(F̃ ) we have sequences (ni ), (q̃i ) with
f˜q̃nii (q̃i + 0 · α) − q̃i
F̃ ni (q̃i ) − q̃i
ρ = lim
= lim
i→∞
i→∞
ni
ni
ni
fq̃nii (0) · α
q̃i + fq̃i (0) · α − q̃i
= lim
∈ ρ(fq̃i ) · α
= lim
i→∞
i→∞
ni
ni
By Corollary 2, ρ(fq̃i ) does not depend upon q̃i so for any p ∈ Td we have ρ ∈ ρ(fp )·α.
Once we ensure that 0 6∈ ρ(fp ) the result follows immediately from [21] (Theorem
1) which states (in part) that ρ(fp ) is either a single point or the convex hull of a
point and zero. If ρ(fp ) is a single point, it must be of the form ρ = ρ(fp ) · α.
Proposition 2 implies that inf p∈Td fp (0) 6= 0. In fact, since displacement is nonnegative, we have inf p∈Td fp (0) − 0 = ε > 0. For any p ∈ T we apply Fact 1 to x = 0
and t = x ∈ R to find
fp (0 + x) − x = fp+xα (0) ≥ ε.
Thus for any sequences (ni ) and (xi ) that realize a ν ∈ ρ(fq̃ )
fpni (xi ) − xi
i→∞
ni
ni −1
fp (fp (xi )) − fpni −1 (xi ) + · · · + fp (xi ) − xi
ni ε
=
≥ lim
= ε,
i→∞ ni
ni
ν = lim
and 0 6∈ ρ(fp ).
2
Proof of Proposition 2:
To show the ( =⇒ ) implication we first find a fixed point of F . By Lemma 2,
p 7→ fp (0) is continuous so the infimum is attained at some p0 ∈ Td , fp0 (0) = 0. We
24
select a p̃0 ∈ π −1 (p0 ) and note that fp0 (0) = 0 ⇐⇒ f˜p̃0 (p̃0 ) = p̃0 ⇐⇒ F̃ (p̃0 ) =
p̃0 ⇐⇒ F (p0 ) = p0 . To show that each fp has non-negative displacement, we note
that inf p∈Td fp (0) = 0 implies that, for any p ∈ Td , x ∈ R, we have fp+xα (0) ≥ 0.
Applying Fact 1 gives fp (x) − x = fp+xα(0) ≥ 0.
In order to show the (⇐=) implication, we note that Corollary 4 gives us that
fp (x) − x ≥ 0 holds for all p ∈ Td and x ∈ R. We again use Fact 1, this time to see
fp (0) = fp−xα(x) − x ≥ 0. As before F (p0 ) = p0 ⇐⇒ fp0 (0) = 0 so inf p∈Td fp (0) = 0.
2
SSM with Fixed Point and Non-Negative Displacement
For the remaining sections of this chapter we will be dealing with SSMs for which
the induced maps fp have non-negative displacement and inf p∈Td fp (0) = 0. As we
have seen these are the SSMs for which the rotation set can be non-trivial and we
focus on them even though some of our easier results could easily be stated in greater
generality.
Fact 2 If F is a SSM with a fixed point and non-negative displacement then, ρ(fp ) =
[0, ρ] for some non-negative constant ρ.
Proof of Fact 2:
First, taking p0 to be a fixed point of F and p̃0 any lift6 of p0 , we note that
6
There is a little subtlety associated with a lift of p0 not having to be fixed, a priori. In general
we have F̃ (p̃0 ) = p̃0 + v for some v ∈ Rd , however since p̃0 and p̃0 + v need both lie on the same
irrationally sloped line we must have v = 0.
25
fpn0 (0) − 0
fp̃n (0) − 0
0−0
= 0
=
= 0,
n
n
n
which places 0 in ρ(fp̃ ).
By using repeatedly that fp (x) − x ≥ 0 for all x ∈ R, we can write
fpn (x) ≥ fpn−1 (x)) ≥ . . . ≥ x.
Thus fpn (x) − x ≥ 0 for all x ∈ R, n ∈ N, forcing ρ(fp ) ⊆ [0, ∞). As ρ(fp ) is also
connected and compact (by Theorem 1), it must be that ρ(fp ) = [0, ρ] for some ρ ≥ 0.
2
When F is a SSM with a fixed point and non-negative displacement we would like
to determine the value of ρ = max ρ(fp ). Particularly, we are interested in when ρ 6= 0.
In order to investigate this, we consider the (displacement) function Φ̃ : Rd → R such
that
F̃ (p̃) − p̃ = Φ̃(p̃) · α,
p̃ ∈ Rd ,
which can be written explicitly as
Φ̃(p̃) :=
1
hF̃ (p̃) − p̃, αi.
||α||2
(2.5)
(Here h· , ·i is the inner product on Rd .)
Note that Lemma 1 secures
Φ̃(p̃ + ṽ) = Φ̃(p̃),
ṽ ∈ Zd , p̃ ∈ Rd ,
so Φ̃ is a lift of a function Φ : Td → R, Φ̃ = Φ ◦ π. We also observe that
Φ̃(p̃) = σp̃−1 (Φ̃(p̃) · α + p̃) = σp̃−1 (F̃ (p̃)) = fp̃ (0).
(2.6)
26
Where the last equality is clear from (2.3) taken with x = t = 0.
We will exploit the following link between the rotation set and invariant measures.
This is a well known connection applicable in a much broader context (see for instance
[8, 22, 21]). It is sometimes used to give an alternative definition of the rotation set
[23, 24].
Lemma 3 For a SSM F with fixed point and non-negative displacement, we have
ν ∈ ρ(fp̃ ) if and only if ν =
R
Td
Φ dµ for some F -invariant probability measure µ.
The argument follows the proof of Theorem 2.4 in [8]. In fact, our lemma is an
immediate consequence of Theorem 2.4 for any map with convex rotation set. We
give a proof for completeness.
Proof of Lemma 3:
By (2.6) we have,
Φ̃(σp̃ (x)) = Φ̃(p̃ + xα) = fp̃+xα (0).
Fact 1 supplies
fp̃+xα (0) = fp̃ (x) − x = (fp̃ − id)(x).
So (fp̃ − id)(x) = Φ̃(σp̃ (x)) and we may define that common value as
Λ(x) :=(fp̃ − id)(x) = Φ̃(σp̃ (x)).
(2.7)
To show the ( =⇒ ) implication, for any ν ∈ ρ(fp̃ ), we will find an F -invariant µ
such that ν =
R
Td
Φ dµ. By the definition of ρ(fp̃ ) there exist some sequences (xk ) ∈ R
27
and (nk ) → ∞ such that, for
nk −1
fp̃nk (xk ) − xk
1 X
νk :=
=
Λ(fp̃i (xk )),
nk
nk i=0
(2.8)
we have νk → ν. We define µ̃k := n1k (δxk + · · · + δf nk −1 (x ) ), so that
p̃
k
Z
nk −1
1 X
i
νk =
Λ(fp̃ (xk )) =
Λ dµ̃k .
nk i=0
R
(2.9)
In order to integrate over a compact set, we consider the push forward of µ̃k under
σp̃ ; namely,
µk :=(σp̃ )∗ (µ̃k ) =
1 δπ(σp̃ (xk )) + · · · + δπ(σp̃ (f nk −1 (x ))) ,
k
p̃
nk
(2.10)
and deduce from (2.7) that
νk =
Z
Λ dµ̃k =
R
Z
Φ dµk .
(2.11)
Td
By (possibly) passing to a subsequence (µki ) we get a weak* limit µ = limi→∞ µki .
In view of π ◦ σp̃ ◦ f = F ◦ π ◦ σp̃ , µk can be also viewed as a normalized measure
distributed on a long F -orbit piece,
µk =
1 δπ(σp̃ (xk )) + · · · + δF nk −1 (π(σp̃ (xk ))) ,
nk
so its limit µ is an F -invariant measure on Td . Thus we have that
ν = lim νki = lim
i→∞
i→∞
Z
Td
Φ dµki =
Z
Φ dµ
Td
as desired.
In order to show the (⇐=) implication, we use the following result.
(2.12)
28
Ergodic Decomposition [25] Let X be a compact metric space and f : X → X be
a measurable map. There exists a set Σ of total measure (i.e. µ(Σc ) = 0 for every
f -invariant measure µ) such that, for any y ∈ Σ, the formula
n−1
1X
Ly (Φ) := lim
Φ(f j (y))
n→∞ n
j=0
(2.13)
defines a bounded linear functional Ly ∈ C(X)∗ and the corresponding measure µy
(such that Ly (Φ) =
R
Φ dµy ) is an ergodic f -invariant measure.
Moreover, given any f -invariant measure µ, for µ-a.e. y, every Φ ∈ L1 (X, µ) is
µy integrable and
Z Z
X
Φ dµy
X
dµ(y) =
Z
Φ dµ.
(2.14)
X
The measure µy (if it exists) is the asymptotic measure of y [26]. The points in y ∈ Σ,
for which µy = µ, are the µ-generic points for µ.
We now show that if µ is any F -invariant probability measure then ν =
ρ(fp̃ ).
R
Td
Φ dµ ∈
Suppose first that µ = µy for some y ∈ Σ. For µy -a.e. q ∈ Td , we can write
q = π(σp̃ (x)) for some p̃ ∈ Rd and x ∈ R and see that
Z
So
n−1
n−1
fp̃n (x) − x
1X
1X
Φ(F i (q)) = lim
Λ(fp̃i (x)) = lim
.
n→∞ n
n→∞
n→∞ n
n
i=0
i=0
Φ dµy = lim
Td
R
Td
Φ dµy ∈ ρ(fp̃ ). By Corollary 2, ρ(fp̃ ) = [0, ρ] so 0 ≤
an arbitrary F -invariant measure, we use (2.14) to write:
Z
Φ dµ =
Td
Z
Td
Z
Φ dµy
Td
dµ(y) ≤
Z
R
Td
Φ dµy ≤ ρ. Now if µ is
ρ dµ = ρ.
Td
(2.15)
(2.16)
29
Since Φ ≥ 0 and by Corollary 2, we then have
R
Td
Φ dµ ∈ [0, ρ] = ρ(fp̃ ).
2
To compute ρ(fp̃ ) we may now use Lemma 3 and (2.16) to observe that
ρ(fp̃ ) ⊆ 0, sup
Z
Φ dµy
Td
⊆ [0, ρ] = ρ(fp̃ )
where the supremum is taken over all F -invariant ergodic measures. By weak* compactness of the space of invariant measures for F , there is a measure η that realizes
that supremum:
ρ=
Z
Φ dη.
(2.17)
Td
We note that if no ergodic measure attained the supremum then the inequality in
(2.16) is strict and we would have
we henceforth do.
R
Td
Φ dη < ρ. So η may be chosen ergodic, which
Flowable SSM with a Stopped Point
Definition 3 A simple solenoidal map F : Td → Td is flowable if and only if there
is a flow (F t )t∈R on Td such that F t is simple solenoidal (with the same non-resonant
α) for all t ∈ R and F is the time-one-map, F = F 1 . Also, if F has a unique fixed
point p0 then we call F a flowable SSM with a single stopped point.
Observe that, for any flowable SSM F with a single stopped point, the difference
fp (x) − x cannot change sign at any p ∈ Td and x ∈ R. By possibly conjugating F
by the central symmetry p 7→ −p, we can always secure fp (x) − x ≥ 0, which we will
assume in all our considerations.
30
We will study the following construction of a flowable SSM, which was introduced into rotation set theory by Katok [21, 27, 28] but had been a known source of
pathological examples in topological dynamics for some time before that (for instance
Example 4.06 in [29]).
We consider a continuous function Ψ : Td → R such that Ψ ≥ 0 and a single
p0 ∈ Td is the unique pre-image of 0 under Ψ. Letting π : Rd → Td be the usual
quotient map, we define Ψ̃ as the lift of Ψ so that the following diagram commutes.
Rd
Ψ̃
π
Td
Ψ
R
Let X be the vector field on Td that lifts to a constant vector field X̃ on Rd given at
p̃ ∈ Rd by
X̃(p̃) = α,
where α = (α1 , . . . , αd ) ∈ Rd is a fixed non-resonant vector. We can assume αd = 1
without losing generality.
At this point we want to require that the vector field Ψ̃ · X̃ has a uniquely defined
continuous flow, (F̃ t )t∈R . This is certainly the case when Ψ is C 1 , which would be
good for most applications. However, in a separate section7 we develop a weaker
necessary and sufficient condition
7
Starting on page 42.
31
(EUH)
Z
0
1
ds
= +∞.
Ψ(p0 ± sα)
(2.18)
We define F̃ : Rd → Rd as the time-one-map of the flow of Ψ̃ · X̃,
F̃ := F̃ 1 .
Such F̃ clearly preserves each line in the direction α and thus is a lift of an SSM
F : Td → Td . Observe that each fp is strictly increasing and that F has a single fixed
point, p0 .
The interesting aspect of this construction is that, even though all flow lines on
Td come arbitrarily close to the stationary point p0 , and do so often, it is possible to
have a situation where some (in fact most) orbits have a positive average speed. This
phenomenon was pointed out by Katok to Handel [28] and has appeared in other
sources such as [27, 21]. The following theorem expounds upon and codifies this idea,
the elements of which are not original, rather they have seemingly only existed in the
related “mathematical folklore” of the subject.
32
Theorem 5 If Ψ is a non-negative function on Td and F is the flowable SSM with
a single stopped point associated to Ψ, then the following are equivalent:
(i) For all p ∈ Td , ρ(fp ) is not a point.
fpn (x) − x
(ii) There is p ∈ T such that for some x ∈ R, lim
exists and is not 0.
n→∞
n
d
fpn (x) − x
exists and is not 0.
n→∞
n
(iii) There is p ∈ Td such that for all x ∈ R, lim
(iv)
Z
[0,1]d
1
|dr̃| < +∞ (integrated according to the ordinary Lebesgue measure).
Ψ̃(r̃)
Moreover, assuming that these conditions hold, for all p ∈ Td we have
" Z
−1 #
1
ρ(fp ) = 0,
|dr̃|
,
[0,1]d Ψ̃(r̃)
(2.19)
and there exists a set A ⊆ Td of full Lebesgue measure such that for all p ∈ A and
x ∈ R the limit in (ii) exists and equals
fpn (x) − x
=
lim
n→∞
n
Furthermore, if
R
[0,1]d
1
Ψ̃(r̃)
Z
[0,1]d
1
|dr̃|
Ψ̃(r̃)
−1
.
|dr̃| = +∞, then ρ(fp ) = {0} and limn→∞
(2.20)
fpn (x)−x
n
= 0 for
all p ∈ Td and x ∈ R. In this case F is uniquely ergodic and the unique invariant
measure is given by δp0 .
The proof of the theorem requires some buildup and will be easily assembled from
several lemmas collecting the main technical insights. Since the rotation set ρ(fp ) (by
Lemma 2) is the same for all p ∈ Td and we want to emphasize the dependence on
Ψ, we shall adopt the notation
ρ(Ψ) :=ρ(fp ).
(2.21)
33
We already know that ρ(Ψ) is of the form [0, ρ] where ρ is given by (2.17). The main
thrust of our arguments is to express the integral in (2.17) in terms of Ψ and α. Our
reliance on the technique of ergodic averaging will require working with fp for which
p is in some full measure “good set”. For convenience, we will work in the universal
cover with the lift p̃ ∈ Rd in the place of p ∈ Td , since letting p = π(p̃) secures fp = fp̃ .
We begin with a description of the ODE governing the flow followed by identification
of the “good” p̃ ∈ Rd .
Let p˜0 ∈ [0, 1)d be the lift of p0 in [0, 1)d, π(p˜0 ) = p0 , and let us consider a fixed
p̃ 6∈ [p̃0 + Zd ]
where [p̃0 + Zd ] =
The function
S
v∈Zd [p̃0
(2.22)
+ v] is the preimage under π of the leaf through p0 .
x(t) := f˜p̃t (x̃) = F̃ t (x̃),
t ∈ R,
is a solution to the initial value problem (IVP)
(
ẋ
= Ψ̃(x)α
x(0) = x̃.
(2.23)
It is convenient to use the parametrization σp̃ of [p̃] to transfer the above IVP
to one on R (with p̃ entering as a parameter). Specifically, the initial condition
x̃ ∈ [p̃] ⊂ Rd corresponds to x = σp̃−1 (x̃) ∈ R, and we can write
x(t) = f˜p̃t (x̃) = x̃ + ∆t · α
where
∆t :=fpt (x) − x.
(2.24)
34
Then
d ˜t
d
d
fp̃ (x̃) = (x̃ + ∆t · α) = ∆t · α,
dt
dt
dt
whereas (2.23) gives
d ˜t
f (x̃) = Ψ̃(f˜p̃t (x̃)) · α = Ψ̃(x̃ + ∆t · α) · α.
dt p̃
Comparing the last two equations gets us the promised one-dimensional IVP:
(
˙t
∆
∆0
= Ψ̃(x̃ + ∆t · α)
= 0.
(2.25)
In the “separated form”,
d∆t
= dt.
Ψ̃(x̃ + ∆t · α)
Now, the requirement p̃ 6∈ [p̃0 + Zd ] guarantees that limt→±∞ ∆t = ±∞ since the
vector field restricted to [p̃] has no rest points, i.e., Ψ̃(x̃ + s · α) 6= 0 for all s ∈ R. It
follows that, for every ∆ ∈ R, there is a unique t ∈ R such that ∆t = ∆. This t is
the time it takes for the solution of (2.25) to travel the distance ∆ (and the solution
of (2.23), in Rd , to travel the distance ∆ · kαk), we shall denote it by t∆ (suppressing
the dependence on the initial point x̃). Clearly,
∆t = ∆ if and only if t∆ = t.
(2.26)
To compute “the time of travel”, we use the substitution y = ∆t and integrate
the “separated form” from t = 0 to t = T (x̃), where T (x̃) is the time it takes to flow
from x̃ to x̃ + α,
Z
0
1
dy
=
Ψ̃(x̃ + y · α)
Z
0
T (x̃)
dt = T (x̃).
(2.27)
35
The time tL of travel from x̃ to x̃ + L · α, where L ∈ N, is then given by
tL = T (x̃) + T (x̃ + α) + · · · + T (x̃ + (L − 1)α) =
L−1
X
T (x̃ + kα).
(2.28)
k=0
Writing x̃ = (x1 , x2 , . . . , xd−1 , xd ), since α is non-resonant and αd = 1, we see that
{x̃ + kα (mod 1) : k ∈ Z} is a dense subset of [0, 1]d−1 × {xd }. In fact, the mapping
x̃ 7→ x̃ + α induces an ergodic translation on the co-dimension one sub-torus
d−1
Td−1
× {xd } /Zd ,
xd := [0, 1]
so whenever T is integrable8 over [0, 1]d−1 the Birkhoff Ergodic Theorem assures that,
−1
for a set of full Lebesgue measure Axd ⊆ Td−1
xd , if x̃ ∈ Ãxd :=π (Axd ), then
L−1
1
lim tL =
L→∞ L
1X
lim
T (x̃ + kα)
L→∞ L
k=0
Z
=
T (x1 , x2 , . . . , xd−1 , xd ) dx1 dx2 . . . dxd−1 .
(2.29)
[0,1]d−1
We now define a full measure set à ⊂ Rd by
à :=
[
Ãxd + Rα
xd ∈R
so that x̃ ∈ Ã if and only if
x̃ = ỹ + aα
(2.30)
for some xd ∈ R, ỹ ∈ Ãxd and a ∈ R. In fact, we may insist that a ≥ 0 (since Ãxd is
invariant under the translation by α).
8
From (2.27) we see that T is in L1 ([0, 1]d−1 ) whenever 1/Ψ̃ is in L1 ([0, 1]d ) so is met concurrently
with condition iv of Theorem 5.
36
Assume that the initial point in the (IVP), x̃ ∈ Ã. This is the same as requiring
that the p̃ we fixed at the outset satisfies p̃ ∈ Ã. (Ã is the set of “good” p̃ we promised
to identify.) Our next goal is to see that (2.29) holds for any x̃ ∈ Ã.
For any L ∈ N, we set N = ⌊L − 1 − a⌋ (where a is as in (2.30)) and compare the
times of travel from ỹ to x̃ + (N + 1)α, from ỹ to ỹ + Lα, and from ỹ to ỹ + (N + 2)α
in order to parlay the obvious inequalities a + N + 1 ≤ L ≤ a + N + 2 into
−t−a +
N
X
k=0
T (x̃ + kα) ≤
L−1
X
k=0
T (ỹ + kα) ≤ −t−a +
N
+1
X
T (x̃ + kα)
k=0
where −t−a is the time of travel9 from ỹ to x̃ (cf. (2.26)). As a consequence, we have
L−1
L−1
k=0
k=0
1X
1X
lim
T (ỹ + kα) = lim
T (x̃ + kα).
L→∞ L
L→∞ L
Thus, as desired, (2.29) indeed extends to all x̃ ∈ Ã:
1
lim tL =
L→∞ L
Z
T (x1 , x2 , . . . , xd−1 , xd ) dx1 dx2 . . . dxd−1 ,
[0,1]d−1
x̃ ∈ Ã.
(2.31)
We are ready to work the integrability of 1/Ψ̃ into the picture:
Lemma 4 If
1
[0,1]d Ψ̃(r̃)
R
|dr̃| < +∞, and the initial point x̃ ∈ Ã, we have
t∆
lim
=
∆→∞ ∆
Z
[0,1]d
1
|dr̃| < +∞.
Ψ̃(r̃)
(2.32)
Proof of Lemma 4:
We first show that it suffices to take the limit along a discrete sequence
t∆
tL
= lim
.
∆→∞ ∆
L→∞ L
lim
(2.33)
9
The awkward minus signs result from the initial point of the journey being ỹ not x̃, as assumed
in the definition of t∆ .
37
To do this we define for each ∆ > 0, L as the smallest positive integer such that
∆ ≤ L, hence tL−1 ≤ t∆ ≤ tL and L − 1 ≤ ∆ ≤ L. Thus
t∆
tL
tL
tL L
tL
≤ lim
≤ lim
= lim
= lim
.
∆→∞ ∆
∆→∞ ∆
L→∞ L − 1
L→∞ L (L − 1)
L→∞ L
lim
Similarly
tL−1 L − 1
L→∞ L − 1
L
tL−1
tL
= lim
,
= lim
L→∞ L − 1
L→∞ L
t∆
tL−1
≥ lim
=
∆→∞ ∆
L→∞ L
lim
lim
proving (2.33). Thus, since x̃ ∈ Ã, (2.31) yields
t∆
tL
lim
= lim
=
∆→∞ ∆
L→∞ L
Z
T (x1 , x2 , . . . , xd−1 , xd ) dx1 dx2 . . . dxd−1 .
[0,1]d−1
(Here xd is fixed. Indeed, x̃ ∈ Ã amounted to x̃ = ỹ + aα with ỹ ∈ Axd for some xd .)
Then applying (2.27) gives
t∆
lim
=
∆→∞ ∆
=
Z
Z[0,1]
d−1
[0,1]d
Z
1
0
1
dy dx1 . . . dxd−1
Ψ̃((x1 , . . . , xd ) + y · α)
1
|dr̃|,
Ψ̃(r̃)
(here we changed the integration variables from (x1 , . . . , xd−1 , y) to (r̃1 , . . . , r̃d ) via
r̃i = xi + yαi ).
2
Here is how the integrability of 1/Ψ̃ can be characterized by dynamics of fp̃ .
38
Lemma 5 For flowable single stopped point SSM the following are equivalent
(i)
Z
[0,1]d
1
|dr̃| < +∞
Ψ̃(r̃)
fp̃n (x) − x
(ii) There is p̃ ∈ R such that for all x ∈ R, lim
exists and is not 0.
n→∞
n
d
fp̃n (x) − x
(iii) There is p̃ ∈ R such that for some x ∈ R, lim
exists and is not 0.
n→∞
n
d
Proof of Lemma 5: (ii) =⇒ (iii) is immediate. To show that (iii) =⇒ (i) we fix
p̃ ∈ Rd and x ∈ R so that υ := limn→∞
fp̃n (x)−x
n
6= 0. Upon setting x̃ := p̃ + x · α, we
can rephrase the definition of υ in terms of ∆t as follows
∆n
∆t
= lim
n→∞ n
t→∞ t
υ = lim
where the second limit taken along t ∈ R is justified by the usual argument based on
the monotonicity of ∆t as a function of t. Putting together (2.28) and (2.33) gives
L−1
1
t
t∆
tL
1X
T (x̃ + kα),
= lim
= lim
= lim
= lim
L→∞ L
L→∞ L
υ t→∞ ∆t ∆→∞ ∆
k=0
(2.34)
where the second equality used t 7→ ∆t being continuous, onto, and limt→∞ ∆t = ∞.
To finish we have to make connection between the time average of T above and
the space integral of 1/Ψ. We did this before for “good” x̃ via Birkhoff Ergodic
Theorem but now we have no control over x̃ forcing the unique ergodicity into play.
We first have to approximate T by continuous functions which we take to be
TN : [0, 1]d → R given by
TN (x̃) :=
T (x̃)
N
if T (x̃) ≤ N
otherwise.
(2.35)
39
The Monotone Convergence Theorem gives
lim
N →∞
Z
TN dx1 . . . dxd−1 =
[0,1]d−1 ×{xd }
Z
T dx1 . . . dxd−1 .
[0,1]d−1 ×{xd }
The unique ergodicity of the translation by α acting on Td−1
xd (where xd is the last
coordinate of our fixed x̃) gives
L−1
1X
lim
TN (x̃ + kα) =
L→∞ L
k=0
Z
TN dx1 . . . dxd−1 .
[0,1]d−1 ×{xd }
First using (2.27) to pass from 1/Ψ̃ to T and then working in the two limits above,
we get
Z
[0,1]d
1
|dr̃| =
Ψ̃(r̃)
=
Z
[0,1]d−1 ×{xd }
Z
Z
1
0
1
dy dx1 . . . dxd−1
Ψ̃((x1 , . . . , xd ) + y · α)
T dx1 . . . dxd−1
[0,1]d−1 ×{xd }
L−1
=
1X
TN (x̃ + kα)
N →∞ L→∞ L
lim lim
k=0
≤
1
L→∞ L
lim
L−1
X
T (x̃ + kα).
k=0
Combining this with (2.34) yields
Z
[0,1]d
1
1
|dr̃| ≤ < +∞
υ
Ψ̃(r̃)
and thus establishes (i).
To show (i) =⇒ (ii) we assume that the integral
1
[0,1]d Ψ̃(r̃)
R
|dr̃| is finite. Fix any
p̃ ∈ Ã. Consider x ∈ R. Since à is invariant under translations in the direction of α,
x̃ = p̃ + x · α is in à as well. Lemma 4 can then be invoked to get
Z
[0,1]d
1
t∆
t
t
|dr̃| = lim
= lim
= lim t
.
t→∞ ∆t
t→∞ f (x) − x
∆→∞ ∆
Ψ̃(r̃)
p̃
40
That is
fp̃n (x) − x
lim
=
n→∞
n
Z
[0,1]d
1
|dr̃|
Ψ̃(r̃)
−1
(2.36)
for all x ∈ R (and p̃ ∈ Ã). In particular, (ii) holds.
2
Finally, we improve on Lemma 5 by explicitly identifying the rotation set.
Corollary 5
1
[0,1]d Ψ̃(r̃)
R
|dr̃| < +∞ if and only if ρ(Ψ) is a non-degenerate segment,
i.e., ρ(Ψ) = [0, ρ] for some ρ 6= 0. Moreover, in such case,
ρ=
Z
[0,1]d
1
|dr̃|
Ψ̃(r̃)
−1
.
(2.37)
Proof of Corollary 5: We start with the (=⇒) implication.
When
1
[0,1]d Ψ̃(r̃)
R
|dr̃| < +∞, (2.36) gives that, for all x ∈ R and all p̃ ∈ Ã,
fp̃n (x) − x
lim
=
n→∞
n
Z
[0,1]d
1
|dr̃|
Ψ̃(r̃)
−1
.
In particular, we have a non-zero element in ρ(Ψ), forcing the right endpoint of ρ(Ψ)
to be non-zero as well, i.e., ρ =
R
Td
Φ dη > 0 in (2.17).
We got that ρ(Ψ) is non-degenerate but we have to identify the ρ, i.e., show (2.37).
The idea is to see that there are lifts of η-generic points in Ã.
The F -invariant measure η can be pushed forward by the flow and averaged to
generate a flow invariant measure ξ,
ξ :=
Z
1
0
F∗t (η) dt.
(2.38)
41
If we restrict attention to the full measure subset B ⊂ Td that is the complement
of the leaf through p0 , then the flow F t has no rest points and the expected return
time to B is positive and finite. Letting b(x) denote the first return to B of a point
x ∈ B we see that the collection of lines {[x, b(x)) | x ∈ B} make up a regular
partition for the flow [30]. So we may view this cross section as the base space of
a special representation of the flow under the first return map. The flow invariant
measure ξ then induces an invariant measure on the base space, Td−1
xd , of the special
representation of the flow [31].
The return map to Td−1
is the familiar translation by α so unique ergodicity
xd
dictates that the induced measure on Td−1
xd is the Lebesgue measure. Because Axd ⊂
Td−1
is of full Lebesgue measure, it follows that, for η-almost every q ∈ B, there
xd
exists p̃ ∈ Ãxd and x ∈ R so that q = π(σp̃ (x)) = π(p̃ + xα). Taking q to be η-generic
so that its asymptotic measure µq = η (cf. (2.15)), we get
Z
[0,1]d
1
|dr̃|
Ψ̃(r̃)
−1
fp̃n (x) − x
= lim
=
n→∞
n
Z
Φ dη = ρ.
(2.39)
Td
It remains to prove the (⇐=) implication.
We assume that ρ(Ψ) is non-degenerate. Its right endpoint is ρ =
R
Again (via (2.15)) taking p̃ ∈ Rd for which q := π(p̃) is η-generic we find
fp̃n (0) − 0
= ρ > 0.
n→∞
n
lim
This verifies (ii) of Lemma 5 and thus secures
Z
[0,1]d
1
|dr̃| < +∞.
Ψ̃(r̃)
Td
Φ dη > 0.
42
2
Proof of Theorem 5:
Throughout we use (2.21), that is ρ(fp̃ ) = ρ(Ψ). Corollary 2 gives us that
(i) is equivalent to ρ(fp̃ ) being a nondegenerate segment, ρ(fp̃ ) = [0, ρ], ρ > 0.
Corollary 5 gives us (i)
⇐⇒
(iv) and secures (2.19). Then Lemma 5 supplies
(ii) ⇐⇒ (iii) ⇐⇒ (iv). To show (2.20), let A := π(Ã) and use (2.36) for p̃ ∈ Ã
that is a lift of p ∈ A.
Suppose now
1
[0,1]d Ψ̃(r̃)
R
|dr̃| = +∞, thus (iv) is not met and the negations of
statements (i), (ii) and (iii) hold. In particular, negating (ii), we have that all p ∈ Td
and x ∈ R are such that limn→∞
fp̃n (x)−x
n
= 0. While the negation of (i) supplies that
ρ(fp ) is a point, taken with Fact 2 (ρ(fp̃ ) = [0, ρ]) we have, ρ(fp̃ ) = {0}. To see
that δp0 is the only ergodic measure we note that, by Lemma 3, for any F -invariant
ergodic measure η,
R
Td
Φ dη = 0. Since Φ ≥ 0, the support of η must be contained
in the set of the x for which Φ(x) = 0. The lifts x̃ of such x satisfy Φ̃(x̃) = 0, i.e.,
F̃ (x̃) − x̃ = 0. Since F̃ is the time one map of Ψ̃ · X̃, it must be that x̃ is stationary,
Ψ̃(x̃) = 0. Hence, x̃ ∈ π −1 (p0 ) and so x = p0 . We have shown that the support of η
consists of the single point p0 , i.e., η = δp0 .
2
Existence and Uniqueness for SSM ODE
Earlier (2.18) we made the following additional assumption on the continuous
43
function Ψ to assure existence and uniqueness of the flow:
(EUH)
Z
0
1
ds
= +∞.
Ψ(p0 ± sα)
(2.40)
Written differently (as integrals with respect to the arc length),
1
= +∞ and
p̃0 +[0,1]α Ψ
Z
that is the area under the graph of
1
Ψ
Z
p̃0 −[0,1]α
1
= +∞,
Ψ
(2.41)
restricted to a segment of [p̃0 ] on either side of p̃0
is infinite. (The limit 1 can be, of course, replaced by any ǫ > 0.) This is exactly the
standard hypothesis guaranteeing existence and uniqueness for the one-dimensional
˙ = Ψ(p0 ± ∆α), which is, however, just a “slice” of our ODE.
ODE ∆
Our goal is to establish the following theorem securing existence of the flow (F t )
and its lift (F̃ t ), as postulated in the previous section.
Theorem 6 Under the above hypothesis the (IVP) given by (2.23) has a unique C 1 smooth solution x(t). It exists for all t ∈ R and depends continuously on the initial
condition x̃ ∈ Rd .
Although this theorem can be obtained from the more general Osgood Criterion (see
for instance page 33 of [32]) we supply the following proof for completeness.
Proof of Theorem 6:
Recall that the (IVP) can be viewed as a family of one-dimensional IVPs. Specifically, the solution is of the form x(t) = x̃ + ∆t α where t 7→ ∆t is a solution to
(
˙ t = Ψ̃(x̃ + ∆t · α),
∆
(2.42)
∆0 = 0.
44
This is dealt with in a standard way familiar from the first ODE course [32] except
that we will have to attend to the issue of dependence on the parameter x̃, so let us
recall the details.
For any t ∈ R, ∆t can be found by solving
Z
0
∆t
ds
= t.
Ψ̃(x̃ + s · α)
As long as x̃ is not in a leaf on which Ψ has a zero, i.e. x̃ 6= [p̃0 ] + Zd , the function
1
:=
Ix̃ (y) :=
x̃+[0,y]α Ψ̃
Z
Z
0
y
ds
Ψ̃(x̃ + s · α)
is C 1 and strictly monotonic with limy→−∞ Ix̃ (y) = −∞ and limy→+∞ Ix̃ (y) = +∞. In
particular, Ix̃ is a homeomorphism R → R and the unique solution ∆t to Ix̃ (∆t ) = t is
given by the inverse homeomorphism ∆t = Ix̃−1 (t). By the Inverse Function Theorem,
∆t depends smoothly on t with
d
∆
dt t
= Ψ̃(∆t ), i.e., it truly satisfies the (IVP) and is
C 1 (as a function of t, not x̃).
Now consider the case when the leaf of x̃ contains a zero of Ψ̃, say x̃ ∈ [p̃0 ]. When
x̃ = p̃0 of course ∆t ≡ 0 is a solution. This solution is unique because, if ∆t 6= 0 for
t 6= 0, then invoking our hypothesis yields a contradiction:
Z
0
∆t
ds
= ±∞ = t.
Ψ̃(p̃0 + s · α)
Suppose now x̃ 6= p̃0 , x̃ = p̃0 + s0 α for some s0 6= 0. Without loss of generality
we may take s0 > 0. Then y 7→ Ix̃ (y) is well defined on (−s0 , ∞) and our hypothesis
guarantees that it maps it homeomorphically onto R. Arguing as before we get a
unique C 1 -solution ∆t .
45
This brings us to the crux: showing that x(t) depends continuously on x̃. Let
then x̃n → x̃, tn → t. We have to prove that ỹn :=xn (tn ) converges to ỹ := x(t).
Consider the line segments (possibly degenerate) in the direction of α swept by the
solutions,
Γn :={xn (τ ) : 0 ≤ τ ≤ tn } and Γ :={x(τ ) : 0 ≤ τ ≤ t}.
Integrating with respect to the arc length, as before, we have
Z
Γn
1
= tn
Ψ̃
and
Z
Γ
1
= t.
Ψ̃
We shall use the following lemma, the proof of which is postponed.
Lemma 6 If Γn are segments in direction α and they satisfy inf n diam(Γn ) > 0 and
dist(Γn , p̃0 + Zd ) → 0, then
1
Γn Ψ̃
R
→ ∞.
To clarify, dist(A, B) := inf{|a − b| : a ∈ A, b ∈ B}.
By the lemma, upon perhaps passing to a subsequence, we have two possibilities:
either diam(Γn ) → 0 or inf n dist(Γn , p̃0 + Zd ) > 0.
Suppose first that diam(Γn ) → 0. Then |x̃n − ỹn | → 0 so that ỹn → x̃. If t = 0,
then ỹ = x̃ and we got what we wanted. If t 6= 0, then |x̃n − ỹn | → 0 is only possible if
Γn converges to a point in p̃0 + Zd , as otherwise tn =
R
Γn
Ψ̃−1 ≤ Const diam(Γn ) → 0.
Thus, for some v ∈ Zd , we have x̃n , ỹn → p̃0 + v and so ỹn → ỹ = x̃ = p̃0 + v.
It remains to consider the case when inf n dist(Γn , p̃0 + Zd ) > 0. Since we already
dealt with the case when diam(Γn ) → 0, we may well assume that there is ǫ > 0 so
that diam(Γn ) > ǫ for all n ∈ N.
46
However, as long as y > 0 is such that dist(x̃ + [0, y]α, p̃0 + Zd ) > ǫ/2 and n is
large, the integrand function in
In (y) :=
Z
x̃n +[0,y]α
1
Ψ̃
is evaluated some distance (say ǫ/3) from the singularity and thus is bounded away
from zero and infinity and converges uniformly to the integrand function in
I(y) :=
1
.
x̃+[0,y]α Ψ̃
Z
As a consequence, over the segment of such y (which include all the y for which
x̃+[0, y]α ∈ Γ), the functions In converge to I uniformly with derivatives. This means
that the equations In (y)−tn = 0 are a small C 1 perturbation of I(y)−t = 0, where the
latter has a unique solution y (so that ỹ = x̃ + yα) at which I ′ (y) 6= 0. It follows that
the solutions yn to In (y) = tn converge to y. As a result, ỹn = x̃n + yn α → x̃ + yα = ỹ.
2
Proof of Lemma 6:
Fix a small ǫ > 0 so that inf n diam(Γn ) > ǫ. Fix δ ∈ (0, ǫ/2). We have that
dist(Γn , p̃0 + v) → 0 for some v ∈ Zd . There is no loss of generality in assuming that
v = 0. By a simple geometric consideration of parallel segments in the plane, we
see that there are subsegments Ln ⊂ Γn such that, in Hausdorff metric, Ln → L :=
p̃0 ± [δ, ǫ/2]α for some choice of sign. For large enough n, the singularity is some
distance from Ln , say dist(Ln , p̃0 + Zd ) > δ/2, which guarantees that
Z
Ln
1
→
Ψ̃
Z
L
1
.
Ψ̃
47
(To be precise, one may first have to rewrite the integrals as suitable integrals over
segments of R and observe that the integrands converge uniformly. We leave the
routine details for the reader.) However, by our hypothesis (2.40) — in which we
can replace the 1 with ǫ/2 — we conclude that
desired conclusion
R
Γn
1
Ψ̃
→ ∞ because
R
Γn
1
Ψ̃
≥
R
1
L Ψ̃
→ ∞ as δ → 0. This gives the
1
.
Ln Ψ̃
R
2
Non-flowable SSM with a Stopped Point
We turn attention to simple solenoidal maps with one stopped point constructed
by prescribing a displacement function Φ instead of the velocity function Ψ as before.
Such maps may fail to be flowable and can exhibit some new dynamical phenomena
not seen in the SSM’s arising from flows.
To show that the class of non-flowable SSM’s is non-empty we construct a simple
example of a SSM that could not arise as the time-one-map of a flow. We then give an
explicit criterion (Theorem 7) for non-triviality of the rotation set arising from SSM’s
constructed from a displacement function with certain nice (smoothness) properties.
The proof of this theorem relies upon bounding the iterates of a point under our SSM
between those of two flowable SSM’s, one slower than our SSM, the other faster. As
such it should come as no surprise that Theorem 7 is similar to Theorem 5.
We start by defining the SSM we will use. Fix a non-resonant α ∈ Rd . Let
Φ : Td → R be a given continuous function such that Φ ≥ 0 and some p0 ∈ Td is the
48
unique pre-image of 0 under Φ. We define the map F and its lift F̃ by
F (p) := p + Φ · α
(2.43)
and
F̃ (p̃) := p̃ + Φ̃ · α,
where Φ̃ is a lift of Φ (and the addition is modulo Zd ). F is seen to be a simple
solenoidal map since, for any p ∈ Rd and s ∈ R, we have
F̃ (p̃ + s · α) = p̃ + s · α + Φ̃(p̃ + s · α) · α
h
i
= p̃ + s + Φ̃(p̃ + s · α) · α ∈ {p̃ + t · α}t∈R .
Example: At this point we note that maps constructed in this way need not be
time-one-maps of flows. One such example can be constructed by starting with the
circle homeomorphism g : T → T given by


2x + 1/2
g(x) = 1/2x + 3/4


x − 1/2
0 ≤ x < 1/6
1/6 ≤ x < 1/2
1/2 ≤ x < 1.
Notice that this map is not flowable. Indeed, if g t (x) were a flow for which g = g 1
then, for any t ∈ R and x ∈ T, we would have g t (g 2 (x)) = g 2 (g t (x)). However, if
we consider t such that g t (0) = 1/6 one can explicitly compute g t (g(g(0))) = 1/6.
Whereas g(g(g t(0))) = 1/3, so g t can not be a flow. Here the existence of the t is
guaranteed by the lack of stationary points for the flow; such points would be fixed
points of g and g has none.
49
1
5
6
0
1
2
1
6
1
2
1
6
1
2
5
6
1
1
2
1
Figure 3. Cartesian graph of g and corresponding interval exchange on T.
Let g̃ : R → R be the lift of g normalized so that g̃(0) = 1/2 and Φ(x1 , x2 ) :=
g̃(x2 ) − x2 . There is a simple solenoidal homeomorphism F0 : T2 → T2 with Φ serving
as its displacement. However, this F0 is fixed point free. To remedy this, we modify
it as follows. Fix a point p0 ∈ T2 . Let L be the segment of the leaf between p0 and
F0 (p0 ) and U be the ǫ-neighborhood of L for some small ǫ > 0, say ǫ = 0.02. It
is not hard to construct an isotopy ht : T2 → T2 , t ∈ [0, 1], that is supported on
U, proceeds along the leaves, starts with h0 = Id, and pulls F0 (p0 ) back to p0 , i.e.,
h1 (F0 (p0 )) = p0 . Now, F := h1 ◦ F0 is a SSM fixing the p0 .
Claim 1 The map F is not flowable.
Proof of claim:
Suppose that F is flowable and F t is the flow. The ǫ in the construction of ht ,
is small enough so that we can find q0 = (x1 , 0) ∈ T2 such that the leaf segment
50
U
p0
q1/2
q4/3
q1/6
q0
q1
q7/6
q5/6
Figure 4. A progression of points on T2 on a “proposed” flow-line that avoids the
slowed region U containing the fixed point p0 . Note that F t (F 2 (q0 )) = q7/6 6= q4/3 =
F 2 (F t (q0 )) illustrating that F is not the time one map of a flow.
K from q0 to F 3 (q0 ) does not intersect U. In this way the flow F t restricted to K
gives F0 restricted to K as its time-one-map. Consider the points qx2 :=F s (q0 ) with
s chosen so that π2 (qx2 ) = x2 . Similar to before, we consider the fixed t such that
F t (q0 ) = q1/6 and compute F t (F 2 (q0 )) = F t (F (q1/2 )) = F t (q1 ) = q7/6 . Whereas,
F 2 (F t (q0 )) = F 2 (q1/6 ) = F (q5/6 ) = q4/3 , so F t can not be a flow.
2
51
Theorem 7 Suppose that Φ : Td → [0, ∞) is a displacement function of a simple
solenoidal map with a stopped point, p0 . Assume that Φ is C 1 and convex on a
neighborhood of p0 . Then the following are equivalent:
(i) For all p ∈ Td , ρ(fp ) is not a point.
fpn (x) − x
(ii) There is p ∈ T such that for some x ∈ R, lim
exists and is not 0.
n→∞
n
d
fpn (x) − x
(iii) There is p ∈ T such that for all x ∈ R, lim
exists and is not 0.
n→∞
n
Z
1
(iv)
|dr̃| < +∞ (integrated according to the ordinary Lebesgue measure).
[0,1]d Φ̃(r̃)
d
This parallels Theorem 5 for flowable maps but steers clear of generalizing its
“moreover part” about ergodic measures. The rest of this chapter is devoted to the
proof of Theorem 7, which is achieved by first approximating the map by certain
flows. We then explore the link between the rotation sets of the time one map of
those flows and the rotation set of our SSM. Finally we show that the integral in
Theorem 7 for our SSM and the integrals in Theorem 5 for the approximating flows
either all converge or all diverge.
Approximating the Map with Flows
As before we let p˜0 ∈ [0, 1)d be the pre-image of p0 , under the quotient map
π : Rd → Td .
For any fixed p̃ ∈ Rd , the restriction of Φ̃ to the line [p̃] yields Φ̃p̃ : R → R defined
52
by
Φ̃p̃ (x) := Φ̃(p̃ + xα).
(2.44)
fp̃ (x) = x + Φ̃p̃ (x).
(2.45)
Note that
Since translating p̃ in the direction of α yields just a translated copy of Φ̃p̃ , we will
happily use only the p̃ in a fixed hyper-plane P transversal to α and passing through
p˜0 .
Let us further assume that F is a homeomorphism. We define the following
functions P × R → R
Ψ̃+
p̃ (s) = sup{Φ̃p̃ (x) : x ≤ s ≤ fp̃ (x)},
(2.46)
Ψ̃−
p̃ (s) = inf{Φ̃p̃ (x) : x ≤ s ≤ fp̃ (x)}.
(2.47)
and
−
Fact 3 Ψ̃+ and Ψ̃− are continuous. Additionally, for any x0 ∈ R, Ψ̃+
p̃ (x0 ) = Ψ̃p̃ (x0 ) =
0 if and only if Φ̃p̃ (x0 ) = 0.
Proof of Fact 3:
We first examine the set {x ≤ s ≤ fp̃ (x)} for any p̃ ∈ P and s ∈ R. Since Φ
was taken to be non-negative, monotonicity of fp̃ implies that yp̃ :=fp̃−1 (s) has the
property yp̃ ≤ s ≤ fp̃ (yp̃ ). Monotonicity of fp̃ (x) also assures that the x ∈ [yp̃ , s] are
the only values to share the property that {x ≤ s ≤ fp̃ (x)}. The assumption that F
53
is a homeomorphism ensures the continuity of fp̃−1 which provides the continuity of
(p, s) 7→ [yp̃ , s] treated as a map for P × R into the space of compact segments in R
(with the obvious topology).
The sup (or inf) is acting over the images under Φ̃p̃ of the compact set [yp̃ , s].
Continuity of Φ̃p̃ would assure that the image is compact as well, in fact that the
+
image is the set [Ψ−
p̃ (s), Ψp̃ (s)]. That Φ̃p̃ is continuous when acting on the space of
compact intervals in R is inherited from the continuity of Φ̃p̃ acting on points. The
sup (similarly inf) acting on compact intervals is also continuous at every interval
[a, b].
+
Thus both Ψ−
p̃ (s) and Ψp̃ (s) are a composition of continuous functions and
thus are continuous themselves.
−
To see that the zeros of Φ̃p̃ , Ψ̃+
p̃ and Ψ̃p̃ coincide we make the observation, by
(2.45), that Φ̃p̃ (x0 ) = 0
⇐⇒
fp̃ (x0 ) = x0 . This is enough by itself to ensure
−
Ψ̃+
p̃ (x0 ) = 0 ⇐⇒ Φ̃p̃ (x0 ) = 0. The Ψ̃p̃ case needs the additional observation (which
was already made in the first paragraph of this proof) that monotonicity of fp̃ prevents
x 6= x0 from having the property that x ≤ x0 ≤ fp̃ (x).
2
If we identify P × R with Rd via (p̃, s) 7→ (p̃ + s · α), then (p̃, s) 7→ Ψ̃±
p̃ (s) become
functions Ψ± : Rd → R. They factor to continuous Ψ± : Td → R that are nonnegative with p0 being the unique pre-image of 0. We can associate then a flowable
single stopped point SSMs, F + and F − , to Ψ+ and Ψ− , respectively. This is assuming
that the corresponding IVPs have unique solutions. For now we will take that as an
54
additional assumption; however, we shall verify it for a broad class of Φ later (see
Corollary 8).
For a fixed p̃ ∈ P , we have one dimensional initial value problems on [p̃],
ẋ+ = Ψ̃+
p̃ (x+ )
x+ (0) = x0
ẋ− = Ψ̃−
p̃ (x− )
x− (0) = x0
(2.48)
fp− : x0 7→ x− (1)
(2.49)
with the corresponding time-one-maps
fp̃+ : x0 7→ x+ (1)
conjugate to the restrictions to [p̃] of F + and F − , respectively.
The following lemma gives the key property of F ± .
Lemma 7 For p̃ ∈ P and x ∈ R, we have
fp̃− (x) ≤ fp̃ (x) ≤ fp̃+ (x).
Proof of Lemma 7:
+
By the definition of Ψ̃+
p̃ , as long as x ≤ s ≤ fp̃ (x), we have 1/Ψ̃p̃ (s) ≤ 1/Φ̃p̃ (x).
Thus
Z
fp̃ (x)
x
ds
≤
+
Ψ̃p̃ (s)
Z
fp̃ (x)
x
ds
fp̃ (x) − x
=
=1=
Φp̃ (x)
Φ̃p̃ (x)
which implies that fp̃ (x) ≤ fp̃+ (x) because the integrand
Z
fp̃+ (x)
ds
Ψ̃+
p̃ (s)
x
ds
Ψ̃+
p̃ (s)
,
is positive for almost
all s. (It vanishes for at most one s.)
−
Likewise, by the definition of Ψ̃−
p̃ , as long as x ≤ s ≤ fp̃ (x), we have 1/Ψ̃p̃ (s) ≥
1/Φ̃p̃ (x). Thus
Z
fp̃ (x)
x
ds
≥
Ψ̃−
(s)
p̃
Z
x
fp̃ (x)
ds
fp̃ (x) − x
=
=1=
Φ
(x)
Φ̃−
(x)
p̃
p̃
Z
fp̃− (x)
x
ds
,
Ψ̃−
(s)
p̃
55
which implies that fp̃ (x) ≥ fp̃− (x).
2
By using the monotonicity of fp̃± , the lemma can be extended to the iterates.
Corollary 6 For p̃ ∈ P , x ∈ R, and n ≥ 0,
(fp̃− )n (x) ≤ fp̃n (x) ≤ (fp̃+ )n (x).
Proof of Corollary 6:
It suffices to verify the induction step from n to n + 1. For the first inequality,
we write
(fp̃− )n+1 (x) = fp̃− ◦ (fp̃− )n (x) ≤ fp̃− ◦ fp̃n (x) ≤ fp̃ ◦ fpn (x) = fp̃n+1 (x),
where we first used the induction hypothesis and the monotonicity of fp̃− and then
invoked the Lemma. For the second inequality, we write
fp̃n+1 (x) = fp̃ ◦ fp̃n (x) ≤ fp̃+ ◦ fp̃n (x) ≤ fp̃+ ◦ (fp+ )n (x) = (fp+ )n+1 (x),
where we first used the Lemma and then invoked the induction hypothesis and the
monotonicity of fp̃+ .
2
Corollary 7 We have
ρ(fp̃− ) ⊆ ρ(fp̃ ) ⊆ ρ(fp̃+ )
so that
56
(i) ρ(fp̃+ ) = {0} =⇒ ρ(fp̃ ) = {0}
(ii) ρ(fp̃− ) 6= {0} =⇒ ρ(fp̃ ) 6= {0}
Proof of Corollary 7:
Fix p̃ ∈ P at will. (i) and (ii) follow from the inclusions if we recall that
all the rotation sets involved are of the form [0, ρ]. For the same reason, to prove
ρ(fp̃− ) ⊂ ρ(Ψ), it suffices to take ν − ∈ ρ(fp̃− ) and find ν ∈ ρ(Ψ) with ν − ≤ ν. There
exist (xi ) ⊂ R and (ni ) ⊂ N with ni → ∞ and
(fp̃− )ni (xi ) − xi
→ ν −.
ni
(2.50)
fp̃ni (xi ) − xi
(fp̃− )ni (xi ) − xi
≥
= νi− .
ni
ni
(2.51)
νi−
:=
By the last corollary,
νi :=
The bounded sequence (νi ) has a convergent subsequence (νij ) and
ni
ν := lim
j→∞
νi−j
fp̃ j (xij ) − xij
≥ ν−
= lim
j→∞
nij
is an element in ρ(fp̃ ). The other inclusion is shown in an analogous way.
2
Regularity Near p0 , and Integrals
The main goal of this subsection is to relate the integrals of 1/Ψ± to the integral
of 1/Φ. To do that, and to ensure existence of F ± , we make the following additional
assumption regarding the regularity of Φ at its critical point p0 .
57
(CRH) There is ε > 0 such that
∂
Φ̃(p̃ + xα)
∂x
exists and is a continuous function of
(p̃, x) provided p̃ ∈ P is in the ǫ-neighborhood of p̃0 and x ∈ (−ǫ, ǫ). Moreover,
for each p̃ ∈ P in the ǫ-neighborhood of p̃0 , the function
(−ǫ, ǫ) ∋ x 7→ Φ̃(p̃ + xα)
either has no critical points or has at most one critical point where it attains
its infimum. If such a critical point exists, we denote it by cp̃ .
Observe that if Φ is C 1 and strictly convex near p0 then (CRH) holds. In particular, Φ that is C 2 with a positive definite Hessian at p0 satisfies (CRH). (We shall
also see in Corollary 8 that (CRH) guarantees existence of F ± .)
We restrict our attention to p̃ ∈ P in the ǫ-neighborhood of p̃0 . As noted in the
proof of Fact 3, the computation of Ψ̃±
p̃ amounts to maximization/minimization of
Φp̃ (x) over x ∈ [fp−1 (s), s], and (CRH) combined with elementary calculus gives
−1
Ψ̃+
p̃ (s) = max{Φ̃p̃ (fp̃ (s)), Φ̃p̃ (s)}
(2.52)
−1
Ψ̃−
p̃ (s) = min{Φ̃p̃ (fp̃ (s)), Φ̃p̃ (s), Φ̃p̃ (cp̃ )}.
(2.53)
and
(Above we use the convention that the term Φ̃p̃ (cp̃ ) is omitted when there are no
critical points between fp−1 (s) and s.)
Our technical goal is to estimate the following non-negative quantity for p̃ ∈ P
in the ǫ-neighborhood of p̃0 ,
ε
1
1
−
ds.
−
Ψ̃+
(s)
−ε Ψ̃p̃ (s)
p̃
Z
(2.54)
58
Note that, when ǫ is small, the integral is guaranteed to be finite for p̃ 6= p̃0 .
Lemma 8 (Key Estimate) For any B > 0, there exists ǫ > 0 such when p̃ ∈ P is
in the ǫ-neighborhood of p̃0
Z
ε
1
1
− +
ds ≤
−
Ψ̃p̃ (s)
−ε Ψ̃p̃ (s)
Z
ε
B
ds + 1.
−ε |Φ̃p̃ (s)|
Proof of Key Estimate:
As to the choice of ǫ we note that the continuity of the derivative stipulated by
(CRH) and the continuity of F −1 ensures that, upon perhaps shrinking the ǫ > 0 a
bit, we get |Φ̃′p̃ (ξ)| ≤ B whenever ξ ∈ [fp̃−1 (s), s], s ∈ (−ǫ, ǫ) and cp̃ ∈ (−ǫ, ǫ).
We start by observing that if cp̃ ≤ s ≤ fp̃ (cp̃ ), then fp̃−1 (s) ≤ cp̃ ≤ s and the
−
minimum in the definition of Ψ̃−
p̃ must be attained at cp̃ so that Ψ̃p̃ (s) = Φ̃p̃ (cp̃ ), and
we can write the following inequality
1
Ψ̃−
p̃ (s)
−
1
Ψ̃+
p̃ (s)
≤
1
Ψ̃−
p̃ (s)
=
1
.
Φ̃p̃ (cp̃ )
(2.55)
On the other hand, when s is not in [cp̃ , fp̃ (cp̃ )], then cp̃ is not in [fp̃−1 (s), s] so the
extrema of Φ̃ are attained at the endpoints, giving
1
1
1
1
−
≤
−
.
−
+
−1
Ψ̃p̃ (s) Ψ̃p̃ (s) Φ̃p̃ (s) Φ̃p̃ (fp̃ (s)) Thus if cp̃ ∈ (−ǫ, ǫ) splitting the integral into three yields
(2.56)
59
ε
1
1
−
ds
−
Ψ̃+
(s)
−ε Ψ̃p̃ (s)
p̃
Z cp̃
Z fp̃ (cp̃ )
Z ε
1
1
1
1
1
1
− +
ds +
− +
ds +
− +
ds
=
−
−
−
Ψ̃p̃ (s)
Ψ̃p̃ (s) Ψ̃p̃ (s)
Ψ̃p̃ (s)
−ε Ψ̃p̃ (s)
cp̃
fp̃ (cp̃ ) Ψ̃p̃ (s)
Z cp̃ Z fp̃ (cp̃ )
Z ε 1
1 1
1
1 ≤
−
ds +
−
ds +
ds
−1 (s))
−1 (s))
Φ̃p̃ (s) Φ̃p̃ (cp̃ )
Φ̃p̃ (s) −ε Φ̃p̃ (f
cp̃
fp̃ (cp̃ ) Φ̃p̃ (f
Z ǫ 1
1 ≤
−
(2.57)
ds + 1
−1 (s))
Φ̃p̃ (s) −ǫ Φ̃p̃ (f
Z
where the 1 comes from the middle contribution (since fp̃ (cp̃ ) = cp + Φ̃(cp̃ )). Of course,
when cp̃ 6∈ (−ǫ, ǫ) the same inequality holds without splitting.
Taking t := fp̃−1 (s), we have
1
Φ̃p̃
(f −1 (s))
−
1
1
1
Φ̃p̃ (t + Φ̃p̃ (t)) − Φ̃p̃ (t)
=
−
=
Φ̃p̃ (s)
Φ̃p̃ (t) Φ̃p̃ (fp̃ (t))
Φ̃p̃ (t)Φ̃p̃ (fp̃ (t))
and the Mean Value Theorem yields
1
1 1
−
max |Φ̃′ (ξ)|.
≤
−1
Φ̃p̃ (fp̃ (s)) Φ̃p̃ (s) |Φ̃p̃ (s)| fp̃−1 (s)≤ξ≤s p̃
(2.58)
Finally, by recalling our choice of ǫ and B, we can forge (2.57) into the desired
inequality,
ε
1
1
− +
ds ≤
−
Ψ̃p̃ (s)
−ε Ψ̃p̃ (s)
Z
Z ε
1
1 B
−
ds+1. (2.59)
ds+1 ≤
−1
Φ̃p̃ (s) −ε Φ̃p̃ (fp̃ (s))
−ε |Φ̃p̃ (s)|
Z
ε
2
Corollary 8 Let I = [0, 1] or I = [−1, 0]. The integrals
R
I
ds
Ψ̃+
p̃0 (s)
and
either both finite or both infinite. Moreover, they are infinite exactly when
infinite, and the integral is infinite when (CRH) holds.
ds
I Ψ̃−
p̃ (s)
R
are
0
ds
I Φ̃p̃ (s)
0
R
is
60
By this corollary, hypothesis (CRH) implies a version of hypothesis (EUH) needed
for the existence of the flows of the vector fields Ψ± α. This places the time-one-maps
F ± on a firm footing.
Proof of Corollary 8:
It suffices to work with I = [0, ǫ] or I = [−ǫ, 0]. In fact, to deal with finite
quantities throughout, let us take a very small δ ∈ (0, ǫ) and integrate over Iδ = [δ, ǫ]
or Iδ = [−ǫ, −δ] . Then, estimating as in the proof above, we get
Z
Iδ
ds
≤
Φ̃p̃0 (s)
Z
Z
ds
ds
1
1
≤
+
− +
ds
−
+
−
Ψ̃p̃0 (s)
Iδ Ψ̃p̃0 (s)
Iδ Ψ̃p̃0 (s)
Iδ Ψ̃p̃0 (s)
Z
Z
ds
ds
≤
+B
+ 1.
+
Iδ Ψ̃p̃0 (s)
Iδ Φ̃p̃0 (s)
Z
(2.60)
Thus,
Z
Iδ
ds
1
≤
1−B
Φ̃p̃0 (s)
Z
ds
1
≤
1−B
Φ̃p̃0 (s)
Z
Iδ
ds
+1
+
Ψ̃p̃0 (s)
!
and taking δ → 0, we get
Z
I
I
!
ds
+1 .
Ψ̃+
p̃0 (s)
(2.61)
Setting δ = 0 in (2.60), we also get
Z
I
ds
≤
+
Ψ̃p̃0 (s)
Z
I
ds
≤
Φ̃p̃0 (s)
Z
I
ds
≤
−
Ψ̃p̃0 (s)
Z
I
ds
+B
+
Ψ̃p̃0 (s)
Z
I
ds
+ 1.
Φ̃p̃0 (s)
(2.62)
Now, from (2.61) and (2.62) it is easy to see that if any one of the three integrals is
finite then so are all of them.
Moreover, under(CRH), Φ̃p̃0 (s) and its first derivative vanish at s = 0 so that
61
|Φ̃p̃0 (s)| ≤ Cs for some constant C > 0 and all small s. This secures
Z
I
ds
≥
Φ̃p̃0 (s)
Z
I
ds
= ∞.
Cs
2
Conclusion of Proof of Theorem 7
Clearly, (iii) =⇒ (ii) =⇒ (i). To see that (i) implies (iii), suppose that the
rotation set is [0, ρ] (for some ρ 6= 0) and let η be the ergodic measure (recall (2.17))
for which the average displacement
have that
R
Φdη = ρ. Taking p ∈ Td to be η-generic, we
n−1
1X
Φ(F k (p)) = ρ.
n→±∞ n
k=0
lim
Thus, via (2.15) and (2.17), we get
fpn (0) − 0
= ρ.
n→±∞
n
lim
This already gives (ii). To get (iii) we use the standard straddling argument as follows.
Given x ∈ R, since ρ 6= 0, there is k ∈ Z, such that
fpk (0) ≤ x ≤ fpk+1(0).
Thus
fpk+n (0) ≤ fpn (x) ≤ fpn+k+1 (0)
so we get what we need for (iii),
fpn (x) − x
lim
= ρ.
n→∞
n
62
Now that we know that (i),(ii),(iii) are equivalent it remains to show the following
two implications. Take B ∈ (0, 1) and ǫ > 0 as given by Lemma 8. Set
Bε (p̃0 ) := {p̃ + xα : |p̃ − p̃0 | < ǫ, |x| < ǫ}.
Then what we would like to show is that
Z
Bε (p̃0 )
1/Φ̃ = +∞ =⇒ ρ(fp̃ ) = {0}
Bε (p̃0 )
1/Φ̃ < +∞ =⇒ ρ(fp̃ ) 6= {0}.
and,
Z
We first assume
R
Bε (p̃0 )
1/Φ̃ = +∞. Then since Ψ̃−
p̃ ≤ Φ̃ we have
Z
Bε (p̃0 )
1
≥
Ψ̃−
p̃
Z
Bε (p̃0 )
1
= +∞.
Φ̃
(2.63)
Setting P ′ := P ∩ Bε (p̃0 ) we use the Tonelli theorem [33] and the key estimate
(about the integral (2.54)) given by Lemma 8, to write
Z
Bε (p̃0 )
1
=
Ψ̃+
p̃
Z Z
ε
Z Z
ε
P′
1
+
−ε Ψ̃p̃ (s)
ds |dp̃|
"
#
1
1
1
=
−
−
ds |dp̃|
Φ̃p̃ (s) Ψ̃+
P ′ −ε Φ̃p̃ (s)
p̃ (s)
!
Z
Z ε
Z ε
1
1
1
≥
ds −
− +
ds |dp̃|
−
Ψ̃p̃ (s)
P′
−ε Φ̃p̃ (s)
−ε Ψ̃p̃ (s)
!
Z
Z
Z ε
1
B
≥
−
ds + 1 |dp̃|
Bε (p̃0 ) Φ̃
P′
−ε |Φ̃p̃ (s)|
Z
Z
1
=
(1 − B) −
1 |dp̃|.
Φ̃
Bε (p̃0 )
P′
The last line equals +∞. By Theorem 5, ρ(fp̃+ ) = {0} forcing ρ(fp̃ ) = {0} by
Corollary 7.
63
Next we assume that
R
Bε (p̃0 )
Z
1/Φ̃ < +∞. Then, since Ψ̃+
p̃ ≥ Φ̃, we have
Bε (p̃0 )
1
≤
Ψ̃+
p̃
Z
Bε (p̃0 )
1
< +∞.
Φ̃
(2.64)
Again, we use the key estimate given by Lemma 8 and the Tonelli theorem, to
write
Z
Bε (p̃0 )
1
=
Ψ̃−
Z Z
P′
ε
1
−
−ε Ψ̃p̃ (s)
ds |dp̃|
"
#
1
1
1
+
− +
ds |dp̃|
=
+
Ψ̃−
(s)
Ψ̃
(s)
P ′ −ε Ψ̃p̃ (s)
p̃
p̃
!
Z
Z ε
Z ε
1
1
1
=
ds +
− +
ds |dp̃|
+
−
Ψ̃p̃ (s)
P′
−ε Ψ̃p̃ (s)
−ε Ψ̃p̃ (s)
!
Z
Z
Z ε
1
B
+
ds + 1 |dp̃|.
≤
+
P′
−ε |Φ̃p̃ (s)|
Bε (p̃0 ) Ψ̃
Z Z
The last line is finite since
ε
1/Ψ̃+
p̃ < +∞,
Bε (p̃0 )
R
R
B/Φ̃ < +∞, and
Bε (p̃0 )
R
P′
1 |dp̃| <
+∞. Thus by Theorem 5, ρ(fp̃− ) 6= {0} forcing ρ(fp̃ ) 6= {0} by Corollary 7.
2
64
CHAPTER 3
INVARIANT MEASURES
We have seen on page 24 of Chapter 2 that the invariant measures for a flowable
SSM F with a single stopped point are closely linked to the rotation set ρ(fp ). When
ρ(fφ ) = {0}, Theorem 5 asserts that there is only one invariant measure, δp0 . When
ρ(fφ ) = [0, ρ] with ρ > 0, we have at least one more ergodic measure η for which
R
Φ dη = ρ (recall (2.17)). The question is whether these are the only two ergodic
measures for F .
In this chapter, we show that the answer depends on the solvability of a cohomological equation over a minimal toral translation and show that coexistence of
uncountably many ergodic measures can occur by arranging this equation to have a
solution. This is a phenomenon that is analogous to the failure of unique ergodicity
for a minimal system as found in the classical example due to Furstenberg ([15, 25])
or some billiard systems ([34, 35]). Indeed, we show that a flowable SSM with a
single stopped point is almost minimal in the sense that its orbits are dense with the
exception of the few that converge to the stopped point (Theorem 10). In particular,
ergodic measures other than δp0 have all of the torus as the support. On the other
hand, when more than one such measure exists, then there are dense measurable invariant subsets A ⊂ Td such that the Lebesgue measure for both A and Ac is positive
65
(Corollary 9).
The most interesting aspect of our coexistence example is that the direction of
the flow is irrational and poorly approximable by rationals – for example it can
have the golden mean slope – a feature associated with the most robust and relevant
aperiodic systems. This is unlike in Furstenberg’s and billiard examples exhibiting fast
approximation by periodic systems and the requisite lacunary arithmetic behavior.
Such features are rather pathological in most applied contexts; they are topologically
robust but measure theoretically meager in the parameter space. On the other hand,
the nature of the slowdown at the stopped point necessary for coexistence of many
ergodic measures suggests that such behavior may be hard to find in less contrived
examples occurring in applications [36, 15, 37]. Therefore, finding a natural context
when our coexistence phenomenon truly comes to the fore is an open problem. At
this point, our example is only in the two-dimensional setting of T2 because we rely
on the good understanding of the irrational circle rotation in terms of the continued
fraction expansion of the rotation angle. It would be interesting to extend the theory
to higher dimensions where the “slowdown” may be less artificial.
Another loose end is the understanding of the situation in the non-flowable case.
For instance we do not know if there are SSM with a single stopped point exhibiting
proper compact invariant sets that are not just the stopped point.
66
Coexistence and Cohomological Equation
Let F be a flowable SSM with a single stopped point. Consider an arbitrary
F -invariant ergodic probability measure µ on Td . It is convenient to single out the
leaf L = π([p̃0 ]) containing the stopped point p0 . L is invariant under F , so µ(L) = 0
or µ(L) = 1.
Suppose first that µ(L) = 1. Then any bounded sub-interval J of L that does not
contain p0 must have measure 0, or else µ(L) = ∞ because there are infinitely many
pairwise disjoint segments in L of the form F n (J) (n ∈ N). Thus µ is concentrated
on p0 and µ = δp0 .
Things can get more interesting if we consider the case when µ(L) = 0, which we
assume from now on. The question we are to address is if µ must be necessarily equal
to a measure η that realizes
R
Φ dη = ρ.
Since µ is carried by the space Td \L on which the flow has no rest points it is
natural to take a cross-section. Specifically, take K ⊂ Td−1 to be the complement of
the countably many points where L pierces the “horizontal” sub-torus Td−1 × {0} of
codimension one, that is
K × {0} = (Td \L) ∩ (Td−1 × {0}).
All points in K ×{0} have an infinitely long trajectory under the flow (F t ) generating
a first return map K × {0} → K × {0}. If we recall that the non-resonant α =
(α1 , . . . , αd ) was normalized so that αd = 1, then we see that the return map is just
67
a restriction to the full Lebesgue measure subset K of the irrational rotation on Td−1
as given by
x 7→ x + β
where
β := (α1 , . . . , αd−1 ).
We define the return time function r : K → [0, ∞) by r(x) = t where t > 0 is the
first return time to K × {0},
F r(x) (x, 0) = (x + β, 0).
Here is the promised criterion for presence of many F -invariant measures on Td \L.
Denote by S1 the unit complex circle,
S1 := {z ∈ C : |z| = 1}.
Theorem 8 The time-one-map F = F 1 has either one or uncountably many ergodic
measures on Td \L. The latter case takes place if and only if there is a non-zero
rational constant κ and a measurable ĥ : Td−1 → S1 such that, for Lebesgue a.e.
x ∈ Td−1 ,
ĥ(x + β) = e2πiκr(x) · ĥ(x).
(3.1)
Let us comment a bit about the equation (3.1), which is an S1 -valued cohomological equation (see [38] for background). If one seeks ĥ in the form
ĥ(x) = e2πih(x)
68
then (3.1) has a solution ĥ iff one can find measurable h : Td−1 → R and integer
valued N : Td−1 → Z so that
h(x + β) − h(x) = κr(x) + N(x).
(3.2)
If one insists that N is constant then (3.2) turns into the R-valued cohomological
equation
h(x + β) h(x)
N
−
= r(x) + .
κ
κ
κ
(3.3)
The study of this equation is a classical subject for smooth or at least continuous
r. The singularity in the r we consider throws us a bit off the beaten path. In the
next section we will solve this equation for some carefully chosen analytic r (with one
singularity) in order to exhibit SSM with a single stopped point that have uncountably
many distinct ergodic measures.
To make one more remark, note that integrating both sides of (3.3) uncovers a
necessary condition for its solvability
0=
Z
r(x)|dx| +
Td−1
N
.
κ
That is, for (3.3) to have a solution for some κ ∈ Q \ {0}, it must be that
ravg :=
Z
r(x)|dx|
Td−1
is rational. We do not know if (3.1) can have solutions for κ ∈ Q \ {0} for which
ravg is irrational1 . The root of the difficulty is in the fact that the process of lifting a
continuous map K → S1 to K → R is not even remotely unique [38].
1
Of course such solutions could not be of the form ĥ(x) = e2πih(x) where h satisfies (3.2) with
the function N (x) constant.
69
Proof of Theorem 8
Let us first build up some infrastructure for the argument. We start by lifting
the flow (F t ) from Td to a flow F̄ t on the space on Td−1 × R which we take to cover
Td = Td−1 × T via
((x1 , . . . , xd−1 ), s) 7→ ((x1 + sα1 , . . . , xd−1 + sαd−1 ), s).
(The addition is modulo 1, of course.) This is a regular covering with the deck
transformation group generated by
D̄ : (x, s) 7→ (x + β, s − 1).
The point of this operation is pure convenience: the “tilt” of the original flow lines
is now entirely absorbed by the deck transformation and the flow F̄ t in Td−1 × R
proceeds along the “vertical” lines (x1 , . . . , xd−1 ) = Const. The flow (F̄ t ) moves with
a variable speed and moves (x, 0) to (x, 1) in time r(x),
F̄ r(x) (x, 0) = (x, 1).
To further simplify things we perform a mapping
ψ :K ×R →K ×R
that sends (x, s) to (x, t) where t is the time it takes for (x, 0) to reach (x, s) under
the flow (F̄ t ), i.e.,
ψ(x, s) = (x, t) iff F̄ t (x, 0) = (x, s).
70
The flow conjugated by ψ,
F̂ t := ψ ◦ F̄ t ◦ ψ −1 ,
moves points up with the constant unit speed,
F̂ t (x, τ ) :=(x, τ + t).
(3.4)
The conjugated deck transformation D̂ := ψ ◦ D̄ ◦ ψ −1 is now given on K × R by
D̂ : (x, τ ) 7→ (x + β, τ − r(x)).
(It incorporates the return time function, which is unbounded so there is no continuous
extension of D̂ to Td−1 × R.)
Recall that the ergodic measure η, pushed forward and averaged over times in
[0, 1], produced an invariant measure ξ for the flow (F t ); see (2.38). This measure lifts
to a measure on Td−1 × R and then pushed forward by ψ produces its counterpart
ˆ which is a measure on K × R. ξˆ is invariant under the flow (F̂ t ) and the deck
ξ,
transformation D̂ and gives mass 1 to the fundamental domain ({(x, τ ) : 0 ≤ τ ≤
r(x)}). The same three properties hold for the normalized Lebesgue measure
|dx| ⊗ dt
.
ravg
The two must be equal:
Lemma 9 Any Borel measure ξˆ on K × R that is invariant under the flow (F̂ t ) and
the deck transformation D̂ and gives mass 1 to the fundamental domain equals
|dx|⊗dt
.
ravg
71
Proof: The invariance of ξˆ under the family of translations (x, τ ) 7→ (x, τ + t), t ∈ R,
implies that ξˆ is of the form ξˆ = ν ⊗ |dτ | for some measure ν on Td−1 . The invariance
of the product measure ν ⊗ |dτ | under D̂ readily implies invariance of ν under the
toral rotation x 7→ x + β. This rotation being uniquely ergodic, ν must be of the form
C|dx| for some C > 0, |dx| being the Lebesgue measure on Td−1 . Thus ξˆ = C|dx| ⊗ dt
and integrating over the fundamental domain gives C = 1/ravg .
2
Remark 1 For Lebesgue a.e. x0 ∈ K we let x1 , . . . , xn−1 denote its first n−1 returns
(to K). Ergodic averaging of r over x 7→ x + β gives
r(x0 ) + . . . + r(xn−1 )
→ ravg .
n
At the same time, n is just the accumulated vertical displacement (as measured in the
“original” Td ) upon flowing from (x0 , 0) to (xn , 0) in time tn := r(x0 ) + . . . + r(xn−1).
Thus the ergodic averaging of the displacement function yields
n
→
r(x0 ) + . . . + r(xn−1 )
Z
Φdη = ρ.
Thus
1
ravg = .
ρ
Descending from the level of Z-coverings, we obtain a convenient model for Td \ L
in the form of
TdD := (K × R)/hD̂i.
72
Naturally, the identification of the complement of L in Td ≃ Td−1 × R/hD̄i with TdD
is that given by the factor ψD of ψ. This identification conjugates the original flow
(F t ) restricted to Td \ L to the flow (FDt ) on TdD that is the factor (by D̂) of the
flow (F̂ t ). (FDt ) is just the special flow under the function x 7→ r(x) built from the
toral translation x 7→ x + β (for background on special flows see [31]). The Lebesgue
measure
|dx|⊗dτ
ravg
|dx|⊗dτ
ravg
descends to an invariant measure for (FDt ), which we also denote by
with little risk of confusion.
The question whether F has an invariant measure on Td \ L or not is equivalent
to the same question regarding the time one map FD := FD1 . From now on, let
us consider an FD -invariant ergodic measure µ. The game is to understand when
µ=
|dx|⊗dτ
ravg
— which we call the “vanilla case” — and when µ 6=
|dx|⊗dτ
ravg
— which we
call the “exotic case”.
For t ∈ R, set µt := (FDt )∗ µ. i.e. µt (A) = µ(FD−t(A)). Each µt is FD -invariant and
ergodic. Furthermore,
ensures that
R1
0
µt dt is an invariant measure for the flow FDt , and Lemma 9
Z
1
µt dt =
0
|dx| ⊗ dτ
.
ravg
To better understand the dependence of µt on t, we define
Eµ := {t ∈ R : µt = µ0 } .
Fact 4 Eµ is a closed subgroup of R.
(3.5)
73
Proof: The set Eµ is closed because it is the inverse image of the closed set {µ0 }
under the map R ∋ t 7→ µt , which is continuous in the weak* topology.
To verify the group property, we check that if t1 , t2 ∈ Eµ then t1 − t2 ∈ Eµ . For
any set A ∈ TdD , by the definition of µt and the flow property of FDt , we have
µt1 −t2 (A) = µ(FD−t1 +t2 (A)) = µ(FD−t1 FDt2 (A)) = µt1 (FDt2 (A)).
Since µt1 = µt2 = µ0 , we have µt1 (FDt2 (A)) = µt2 (FDt2 (A)). Thus,
µt1 −t2 (A) = µt2 (FDt2 (A)) = µ(FD−t2 FDt2 (A)) = µ(A) = µ0 (A).
So t1 − t2 ∈ Eµ .
2
If Eµ = R then µt = µ0 for all t ∈ [0, 1], so
|dx| ⊗ dτ
=
ravg
Z
1
µt dt =
0
Z
1
µ dt = µ.
0
That is Eµ = R exactly in the “vanilla situation” when µ =
|dx|⊗dτ
.
ravg
Conclusion of Proof of Theorem 8:
We start with the forward implication ( =⇒ ). That is we assume that F has more
than one invariant ergodic measure on Td \ L and work to establish the cohomological
equation.
By our hypothesis, there is an ergodic measure µ on TdD for which Eµ 6= R. By
the classification of closed subgroups of R [39] and because Z ⊂ Eµ , there is b ∈ N
such that
Eµ =
1
· Z.
b
(3.6)
74
It remains to construct the function ĥ so that (3.1) holds.
Fact 5 µt = µs if and only if t − s ∈ Eµ .
Proof of Fact 5:
Suppose first that µt = µs . For A ⊆ TdD we have
µt−s (A) = µ(FD−t+s (A)) = µ(FD−t FDs (A)) = µt (FDs (A))
= µs (FDs (A)) = µ(FD−s FDs (A)) = µ(A)
so t − s ∈ Eµ .
Suppose now that t − s ∈ Eµ . For A ⊆ TdD we have
µs (A) = µ0 (FD−s (A)) = µt−s (FD−s (A)) = µ(FD−t+s FD−s (A)) = µ(FD−t (A)) = µt (A).
Therefore, µt = µs .
2
Let us now denote by F the family of functions on TdD that correspond to the
continuous functions on the original torus Td . Recalling that Td ≃ Td−1 × R/hD̄i is
identified with TdD via ψD , the precise definition of F is
−1
F := {g ◦ ψD
: TdD → R : g is a continuous function on Td−1 × R/hD̄i}.
−1
We note that F consists of continuous functions on TdD — because ψD
is continuous
— but there are many continuous functions on the noncompact space TdD that are
not included in F . The purpose of introducing F is to have a manageable class of
75
“test functions” such that the measures on TdD coming from the measures on Td \ L
are uniquely determined by integration against the functions in F . The important
point is that, by implementing the Ergodic Decomposition Theorem on the compact
space Td , we can speak of the set Gt of µt -generic points defined by
(
n−1
Gt := x ∈
TdD
| ∀g∈F
1X
g̃(x) := lim
g FDj (x) =
n→∞ n
j=0
Z
TdD
g dµt
)
,
and µt (Gt ) = 1 for all t ∈ R.
Let
G :=
[
Gt =
t∈R
[
Gt
t∈[0, 1b )
where the second equality follows from (3.6) and Fact 5.
Fact 6 For t, τ ∈ R, FDτ (Gt ) = Gt+τ .
Proof of Fact 6:
For any fixed t, τ ∈ R, for all x ∈ Gt and any g ∈ F we have:
n−1
n−1
1X
lim
g ◦ FDτ FDj (x)
n→∞ n
j=0
Z
Z
τ
=
g ◦ FD (x) dµt =
g dµt+τ .
1X
lim
g FDj (FDτ (x)) =
n→∞ n
j=0
TdD
TdD
For the second equality we used that g ◦ FDτ ∈ F , which is quite clear from the
definition of F . So we have that FDτ (Gt ) ⊂ Gt+τ . From the prior inclusion it is
immediate that Gt+k ⊂ FDτ (Gt ) since we have that FD−τ (Gt+τ ) ⊂ Gt+τ −τ = Gt .
2
76
Fact 7 G is a disjoint union of the Gt taken over t ∈ [0, 1b ), has full |dx|⊗dτ measure,
and is flow invariant.
Proof of Fact 7:
Flow invariance is immediate from Fact 6. We compute
!
Z 1
Z 1
[
|dx| ⊗ dτ (G) = ravg
µt
Gτ dt ≥ ravg
µt (Gt ) dt = ravg ,
0
0
τ ∈R
so G is of full |dx| ⊗ dτ measure. To see the disjointness of the Gt ’s observe that
for 0 < t1 < t2 < 1b , t2 − t1 6∈ Eµ . Thus, µt1 6= µt2 by Fact 5. Since both µti come
from FD -invariant measures on Td \ L there exists g ∈ F , where for any x1 ∈ Gt1 and
x2 ∈ Gt2 , we have that
g̃(x1 ) =
Z
g dµt1 6=
Z
g dµt2 = g̃(x2 ).
So x1 6= x2 and Gt1 ∩ Gt2 = {∅}.
2
In view of the last fact we have an |dx| ⊗ dτ -a.e. defined function Υ : TD → S1
given on p ∈ G by
Υ(p) :=ei2πbt iff p ∈ Gt .
(3.7)
Lemma 10 The function Υ is measurable.
Proof of Lemma 10:
One has to show that the preimage under Υ of an arc Jt1 ,t2 between e2πit1 and
e2πit2 is measurable for any t1 < t2 . It is straightforward that, for g ∈ F ,
(Z
)
Gg :=
TdD
g dµt | t ∈ (t1 , t2 )
77
is a Borel set (by weak* continuity of t 7→ µt ) and so is
(
n−1
1X
Xg := p ∈ TdD | lim
g FDj (p) exists and belongs to Gg
n→∞ n
j=0
)
.
Since
Υ−1 (Jt1 ,t2 ) =
[
Xg
g∈F
where the intersection can be taken over a countable dense subset of F , it is also
Borel.
2
Recall [40] that Υ : Td → T is called an eigenfunction of the flow F t : Td ×R → Td
with eigenvalue b if for all t ∈ R and almost all p ∈ Td , Υ(F t (p)) = Υ(p)ei2πbt .
Fact 8 Υ is an eigenfunction of the flow (FDt ) with the eigenvalue b.
Proof of Fact 8:
Fix τ ∈ R and p ∈ G. Take t so that p ∈ Gt . By Fact 6, we have FDτ (p) ∈ Gt+τ .
Thus,
Υ(FDτ (p)) = e2πib(t+τ ) = e2πibτ e2πibt = Υ(p)ei2πbτ .
2
Passing from an eigenfunction to a cohomological equation is then done in a
routine way [38] by restricting attention to the points p ∈ TdD that belong to the
cross-section K × {0}. Here it is important that |dx|-a.e. such points p belong to G
because G is of |dx| ⊗ dt-full measure and invariant under the flow. We have then a
78
|dx|-a.e. defined measurable ĥ : K → T given by
ĥ(x) := Υ(x, 0).
In view of
r(x)
FD (x, 0) = (x + β, 0),
Fact 8 gives
r(x)
ĥ(x + β) = Υ(FD (x, 0)) = e2πibr(x) Υ(x, 0) = e2πibr(x) ĥ(x).
This is the cohomological equation (3.1) so the forward implication of the theorem is
shown.
To show the reverse implication ( ⇐= ) we now assume that, for some κ = p/q,
where p ∈ Z, q ∈ N, with GCD(p, q) = 1, there exists a |dx|-measurable function
ĥ : K → T such that the cohomological equation (3.1) holds. Following the well
known relation between cohomological equations and eigenfunctions for the special
flow, we define Υ : K × R → T by
Υ(x, t) = e2πiκt ĥ(x).
Lemma 11 The function Υ is deck transformation invariant, Υ ◦ D̂ = Υ, and the
quotient of Υ, ΥD : TdD → T is an eigenfunction for FDτ with the eigenvalue κ.
Proof of Lemma 11: To verify the deck invariance, for x ∈ K, τ ∈ R, we invoke
79
the cohomological equation to get
Υ(x + β, τ − r(x)) = e2πiκ(τ −r(x)) ĥ(x + β)
= e2πiκτ e−2πiκr(x) ĥ(x + β)
= e2πiκτ ĥ(x) = Υ(x, τ ).
Thus Υ has a well defined quotient TdD → T. That the quotient is an eigenfunction
is immediate from the definition of Υ since, for x ∈ K, τ ∈ R,
Υ(x, t + τ ) = e2πiκτ Υ(x, t).
2
Returning to the time one map FD = FD1 , we see that e2πiκ is its eigenvalue with
eigenfunction ΥD , i.e.,
ΥD ◦ FD = e2πiκ ΥD .
Eigenvalues of a map form a subgroup of the complex circle T [41], so from κ = p/q,
FD has an eigenvalue 1 with the corresponding eigenfunction, the q-th power of ΥD ,
being an invariant function:
ΥqD ◦ FD = ΥqD .
Note that ΥqD is non-constant because Υ takes on all the uncountably many values in
R/ 1b Z. This implies that FD has non-trivial invariant subsets and thus many invariant
measures, finishing the proof; although, for the reader’s convenience we now proceed
to give the explicit construction of the invariant sets.
2
80
To complete the picture we explicitly generate an FD invariant set S whose
measure is between 0 and 1. Note that viewed in the original Td ,
|dx|⊗dτ
ravg
|dx|⊗dτ
ravg
is a measure
that has positive (integrable) density with respect to the ordinary Lebesgue measure
on Td so that the sets S we produce are FD invariant with Lebesgue measure between
0 and 1. For future reference we state this as a stand-alone corollary:
Corollary 9 If a flowable SSM with a stopped point has more than one non-atomic
invariant measure then it is not ergodic with respect to the Lebesgue measure, i.e.,
there are invariant subsets of Lebesgue measure between 0 and 1.
Proof of Corollary 9:
The construction of S is easy, we just take the preimage under ΥD of a small
arc in S1 . Specifically, for 0 ≤ a < b ≤ 1 define,
Sa,b = {(x, t)|2πa ≤ Arg(ΥqD (x, t)) < 2πb}
where Arg is the branch of the arg that take values in [0, 2π). (The sets depicted
in Figure 2 on page 10 are such sets.) Since ΥqD (FD (x, t)) = Υq (x, t), we have
Arg(ΥqD (x, t)) = Arg(ΥqD (FD (x, t))) and Sa,b is FD invariant.
Fix ε with 0 < ε < 1 and consider S :=S0,ε . For every (x, t) ∈ S there is an m ∈ Z
such that,
Arg(e2πitq (ĥ(x))q ) = 2πtq − 2πm + q Arg(ĥ(x)).
So, for any x ∈ K, m ∈ Z, we have (x, t) ∈ S if and only if
!
"
Arg(
ĥ(x))
m
Arg(
ĥ(x))
ε
m
−
, −
+
.
t ∈ Sxm :=
q
2π
q
2π
q
81
Thus we may observe the following: the length satisfies |Sxm | = ε/q and Sxm = Sxn
if and only if m ∈ n + q · Z. Since K has full Lebesgue measure in Td−1 , Fubini’s
theorem yields
|dx| ⊗ dτ
0<
(S) = q
ravg
Z
K
|Sxm | dx = q · 1 ·
ε
= ε < 1.
q
2
Existence Examples
Theorem 8 tells us exactly when more than one ergodic measure for F exists on
Td \L, but it is not clear that there exist κ ∈ Q and flows with return maps r(x) that
satisfy (3.1).
Theorem 9 There exists a flowable SSM F with a single stopped point for which
there are uncountably many distinct ergodic invariant measures µ.
Cohomology with a constant requires a rather discontinuous ĥ(x), as the return
map of a flow with a stopped point has a (single) singularity at some point x0 .
Proposition 3 If x0 is the only discontinuous point of ei2πκr(x) , then ĥ satisfying
(3.1) must have a dense set of discontinuities.
Proof of Proposition 3:
We are assuming ei2πκr(x) is only discontinuous at x0 . Let D be the set of
discontinuous points of ĥ. From (3.1), we have x0 ∈ D or x0 + β ∈ D. Suppose
x0 ∈ D. Since rewriting (3.1) gives ĥ(x − β) = e−i2πκr(x−β) ĥ(x), it must be that
82
x0 − β ∈ D. (This used that e−i2πκr(x−β) is continuous at x0 .) Replacing x by x − β
etc. in succession yields x0 − 2β, x0 − 3β, . . . ∈ D, that is {x0 + nβ | n ≤ 0} ⊆ D.
In the other case, when x0 + β ∈ D, ĥ(x + 2β) = ei2πκr(x+β) ĥ(x + β) gives
that x0 + 2β ∈ D. As before, we further get x0 + 3β, x0 + 4β, . . . ∈ D, that is
{x0 + nβ | n > 0} ⊆ D.
2
We give a class of examples of flows on T2 for which we can find an appropriate
ĥ so that the return map is cohomologous with a constant. Once we have a pair of
such functions in hand it is easy to adjust the constant to be any value we desire, in
particular we may adjust it so that κ ∈ Q \ {0}.
Our plan is to first show that, for any irrational β with bounded continued fraction
expansion, there exist two functions R and H analytic on D = {z : |z| < 1} that
have the following properties:
(i) Letting R(z) =
∞
X
n=0
by
Rn z n be the power series for R, the function r : T → R given
r(x) := Re
∞
X
n=0
Rn ei2πnx
!
(3.8)
is continuous at all x ∈ T \ {0} and limx→0 r(x) = ∞.
(ii) Letting H(z) =
∞
X
n=0
given by
Hn z n be the power series for H, the function h : T → R
h(x) := Re
∞
X
n=0
Hn ei2πnx
!
(3.9)
83
is a square integrable function on T.
(iii) For a.e. x ∈ T, we have
r(x) = h(x + β) − h(x) +
Z
r(x)|dx|.
(3.10)
T
Looking at (iii), we see that by multiplying r by a constant one can make R0 =
R
T
r(x)|dx| a positive rational and satisfy (3.3) for a suitable κ ∈ Q \ {0}. Then it
still remains to construct a suitable function Ψ so that the flowable SSM constructed
from Ψ has the return function r. That is the least interesting part and we leave it
for last (see Lemma 13).
Plugging the series for r and h into (3.10), comparing the coefficients at the
corresponding terms yields
Rn ei2πnx = Hn ei2πn(x+β) − Hn ei2πnx = Hn ei2πnx ei2πnβ − 1 ,
for n ≥ 1. So, for n ≥ 1, Hn is determined by Rn according to
Hn =
Rn
i2πnβ
e
−
1
.
(3.11)
H0 is immaterial and we may well default it to H0 = 0. We note that (3.11) and its
notorious “small denominator” have a long history, see [2, 25].
Now, let us fix a > 0, b > 2 and 1/2 < p < 1 and set
R(z) := a log
b
1−z
1−p
(3.12)
where we use the standard branches for the logarithm and the power with a cut along
the negative real axis, log(reiθ ) = log r + iθ, (reiθ )1−p = r 1−p ei(1−p)θ with r > 0,
84
θ ∈ (−π, π). This makes R analytic on D as well as a slightly larger open set
containing D \ {1}. We briefly note the importance of the constants. The parameter
a is just used to adjust the integral of r(x) to any desired rational. In choosing b > 2
we not only guarantee that when |z| < 1 the log has no zeros but also that r(x) > 0,
which is a desirable condition when we later construct the SSM that has this return
time r(x). Finally the range of acceptable p-values is chosen to keep h(x) sufficiently
“nice” and is critical for validity of the key Lemma 12.
Our goal is to show that this R together with the H determined by (3.11) satisfy
(i), (ii), and (iii).
We observed that R is analytic on D and has a single singularity on ∂D at z = 1.
In fact, by using ez = 1 + z + . . ., we obtain the asymptotics at z = e2πix = 1 given
by
2πix
R(e
) ≈ a log
b
2πix
1−p
1−p
b
= a log
− log |x|
≈ a (− log |x|)1−p
2πi
(3.13)
The secret to our choice of R lies in the following asymptotics of the Taylor
coefficients taken from [42] [Vol 1, Chapter 5 Theorem 2.31 pg 192]2 :
lim n→∞
Rn
1−p
n(log n)p
= 1.
In particular, there are N and C1 > 0 so that, for n > N, we have
Rn <
2
With α = −1, β = 1 − p.
C1
.
n(log n)p
(3.14)
85
This is what will enable us to show the following lemma, which is the crux of the
whole development.
Lemma 12
X
n>N
|Hn |2 is finite.
The lemma ensures that the Taylor series for H restricted to S1 is convergent in
L2 (S1 ). Hence, H|S1 is square integrable and so is its real part, h. That H is analytic
on D follows, and (3.10) is a.e. satisfied by the construction of R and H.
Proof of Lemma 12:
We may as well take β ∈ (1/2, 1). The argument critically depends on the theory
of continued fractions and its connection with the rotation of T by β. While relegating
a detailed account of this material to an appendix, let us summarize what is needed.
Recall that the continued fraction expansion of β,
1
β=
,
1
a1 +
a2 +
1
..
.
is called bounded if the sequence (ai ) of non-negative integers is bounded. As usual,
we write the partial quotients as reduced fractions pi /qi ,
pi
=
qi
1
(i ∈ N).
1
a1 +
a2 +
1
..
Setting additionally p−1 = 1, p0 = 0, q−1
1
ai
= 0 and q0 = 1, yields the familiar [43]
.+
recurrence relations
pi+1 = ai+1 pi + pi−1
and
qi+1 = ai+1 qi + qi−1 .
86
We shall use the following standard fact (the proof of which is recounted in the
appendix).
Fact 9 If qi are the denominators of the convergents of some number β, then for
i≥2
qi ≥ 2
i−1
2
.
We will also need the Ostrowski β-numeration [44, 45] of m ∈ Z+ ,
m=
∞
X
dk qk ,
k=1
where dk ∈ {1, . . . , ak+1 }, called the digits, are determined by a “greedy algorithm”
(described in the appendix). We shall write
m = [d1 d2 d3 . . . ]
to indicate that the sequence (dk ) is the Ostrowski β-numeration of m.
The following lower bound on the small denominators — also shown in the appendix — will play a pivotal role.
Fact 10 Let
C2 :=
1
.
maxi ai + 2
If n = [0 . . . 0dj dj+1 . . . ] with dj > 0, then
|ei2πnβ − 1| > C2 /qj+1 .
87
Denote by Nj all natural n with the β-expansion of the form [0 . . . 0dj dj+1 . . . ]
with dj > 0. These Nj are pairwise disjoint and their union is N. By using the
asymptotics for Rn , we estimate
X
n>N
|Hn |
2
2
2
X
X
Rn
1
C1 =
ei2πnβ − 1 <
ei2πnβ − 1 n(log n)p n>N
n>N
2
∞ X X
1
C1
≤
ei2πmβ − 1 m(log m)p .
j=1 m∈N
j
By invoking Fact 10 we can further estimate
2
∞ X ∞ X
X
X
1
C1
<
ei2πmβ − 1 m(log m)p j=1 m∈N
j=1 m∈N
2
qj+1
C1
C2 m(log m)p j
2
∞ X X
qj
C1
≤
C 2 m(log m)p 2
j=1 m∈N
j
j
where the last inequality used qj+1 ≤ qj /C2 , which, in turn, follows from
qj+1 = aj+1 qj + qj−1 ≤ aj+1 qj + qj < (aj+1 + 2)qj ≤
qj
.
C2
(3.15)
For each j we order Nj and write it as Nj = {m1 , m2 , m3 , . . . } where ml < ml+1 .
We note that m1 = qj and ml+1 ≥ ml + qj by the nature of the β-expansion. Thus,
ml ≥ lqj allowing us to push the estimate a bit further:
2
2
∞ X ∞ X
∞ X
X
C1 qj
C1 qj
≤
C 2 m(log m)p C 2 lqj (log(lqj ))p .
j=1 m∈Nj
2
j=1 l=1
2
Making use of Fact 9 provides
∞ X
∞ ∞ X
∞ C2 X
2
X
C
1
1
1
1
√
|Hn |2 ≤
=
.
j−1
4
2
l
2p
42
C2 j=1 l=1 l (log( √ ) + j log 2) 2 ))2p j=1 l=1 C2 l (log(l2
n>N
2
X
88
To see that the right hand sum is finite, it is convenient to take l0 ∈ N so that,
√
for l > l0 (= 2), log(l/ 2) ≥ 0 and we have the majorization:
∞ X
∞ X
X
X
1
1
1 1
1
√
√
2
<
2
2p
l
2p
l (log( √ ) + j log 2) l j (log 2)2p
j=1 l>l0
j=1 l>l0
2
=
(log
1
√
2)2p
∞
X
1 X1
.
j 2p l>l l2
j=1
0
Since p > 1/2 the right hand side is finite. For the finitely many l with l ≤ l0 , the
sum over j is also finite by a comparison with
that
P
n>N
|Hn |2 < ∞, as desired.
P∞
1
j=1 j 2p .
Therefore, we have shown
2
The proof of Theorem 9 will be complete once we establish the following.
Lemma 13 Given a non-resonant α ∈ R2 and r : T \ {0} → (0, ∞) such that 1/r
extended by 1/r(0) := 0 is continuous, there exists Ψ : T2 → [0, ∞) with the only
preimage of 0 being p0 = 0 such that the regularity hypothesis (EUH) is satisfied and
the flow (F t ) of the vector field Ψα generates the return function to T × {1/2} equal
to r.
Above, we took the cross-section along T × {1/2} and not T × {0} because we place
the stopped point at (0, 0). Had the stopped point sat in the cross-section, the return
time would have two points of discontinuity where it converges to infinity, at (0, 0)
and (−α, 0) ∈ T × {0}. With our choice of a shifted cross-section, there is only one
discontinuity point, at (−α/2, 1/2) ∈ T × {1/2}. However, in the argument below,
89
we find it convenient to parameterize T × {1/2} with x ∈ T so that x corresponds to
(x − α/2, 1/2) and the discontinuity appears at x = 0.
Sketch of Proof: Again it is more convenient to work in the context of the torus
T × R/hD̄i where the (x, s) is a local coordinate and the flow is vertical. Also, it
is clear that the difficulty is in realizing the prescribed return time for the points
near the stopped point so we will concentrate attention on a square ǫ neighborhood
of (0, 0) for some small ǫ > 0. In fact, let us take a bit less than the half of the
available return time, r1 (x) := (r(x) − C)/2, where C ∈ (0, minx r(x)) and construct
Ψ̄(x, s) for (x, s) ∈ (−ǫ, ǫ) × (0, ǫ) so that the travel time through this upper half of
the neighborhood is
Z
ǫ
0
ds
= r1 (x) (|x| < ǫ).
Ψ̄(x, s)
This locally defined Ψ̄ may now be extended to a desired global function, first by
symmetry to realize the r(x) − C portion of the return time, and then by routine
smooth extension to use the rest of the remaining time C.
One can seek Ψ̄ in many different forms, we choose to take
Ψ̄(x, s) := s2 + (ψ(x))2
where ψ is a non-negative function with ψ(0) = 0 to be determined. Note that as
soon as ψ is continuous, Ψ̄ of this form is continuous and satisfies (EUH) because
Rǫ
ds
0 s2
= +∞.
90
We have
Z
ǫ
0
ds
=
Ψ̄(x, s)
Z
ǫ
0
ds
= arctan
2
s + (ψ(x))2
ǫ
ψ(x)
1
π
≈
ψ(x)
2ψ(x)
where the asymptotics is for |x| → 0. Because u 7→ arctan (ǫu) u is a homeomorphism
of [0, ∞) to itself, solving
arctan
for u =
1
ψ(x)
ǫ
ψ(x)
1
= r1 (x)
ψ(x)
and then ψ(x) itself represents no problem and yields ψ(x) that is
continuous at x 6= 0 and has the asymptotics near x = 0 given by
ψ(x) ∼ 2
π
π
=
,
2r(x)
r(x)
where the leading 2 is due to the same slowdown in the lower portion. This makes ψ
continuous at 0 and also otherwise exactly as desired.
2
Minimality (Almost)
Recall that a homeomorphism F of a compact space is called minimal if and only
if there are no proper closed non-empty subsets A that are invariant under F in the
sense that A = F (A). The following theorem shows that flowable SSM with a stopped
point come pretty close to being minimal, even though they may have uncountably
many distinct ergodic measures when the cohomological equation (3.1) in Theorem 8
has solutions.
91
Theorem 10 For a flowable SSM F with a single stopped point p0 , the only closed
invariant sets are {p0 } and Td . In particular, any F -orbit in Td \ L is dense in all of
Td .
Because any closed invariant set carries an ergodic invariant measure, the theorem
follows once we see that the support of a F -invariant ergodic measure is either {p0 }
or Td . In fact, δp0 being the only atomic invariant measure, it suffices to establish the
following proposition.
Proposition 4 The support of a non-atomic ergodic measure µ is all of Td .
Lemma 14 Any non-empty open U ⊂ Td that is F -invariant has cl(U) = Td . That
is U, is dense in Td .
This lemma by itself implies topological transitivity, i.e. existence of a dense
orbit.
Proof of Lemma 14:
Define L− as the set of points that flow in forward time to the stopped point p0 ,
and L+ as the set of points that flow to p0 in backwards time. We define Ll ⊂ L+
as the finite subarc starting at p0 and of length l. Taking as before p̃0 ∈ Rd so that
π(p̃0 ) = p0 we denote by L̃l the line segment obtained as the lift of Ll that contains
p̃0 . (We are lifting with respect to π : Rd → Td .)
Fix an arbitrary ∆ > 0 and choose l such that Ll is ∆/2-dense on Td . From
irrationality of the flow we can find an ε, with 0 < ε < ∆/2, where the ε neighborhood
of L̃l intersects p̃0 + Zd only at p̃0 .
92
Un
p0
U0
Figure 5. A picture proof of Lemma 14, showing U0 , and Un = F n (U0 ) whose ”tails”
grow (in length) with n; each is contained in the F -invariant set U.
Since L− is dense, U contains a point p− ∈ L− . As F (U) = U, F m (p− ) ∈ U
for all m ∈ N. We select a large enough m, so that pε :=F m (p− ) ∈ U is in the εneighborhood of p0 . Fix a small enough δε so 0 < δε < ε and U0 := Bδε (pε ) ⊂ U. For
large n we observe that the open set Un := F n (U0 ) ⊂ U is a topological disk, shaped
like a very elongated letter U whose bend winds tightly against the fixed point p0 and
the arms stretch far along L+ . Figuratively speaking, as it is pushed by the flow, U0
get ”snagged” on p0 . The idea is that, for sufficiently large n, the arms (and thus U)
are ∆-dense in Td . By arbitrariness of ∆ > 0, U is dense in Td .
For the reader that is not yet convinced we argue this point a bit more carefully.
We select p ∈ U0 \ L− and a continuous path γ ⊂ U0 from pε to p. Since F n (pε )
cannot advance along L past p0 and p ∈
/ L− is displaced by the flow by an amount
that increases to infinity with increasing time, we may find an n so that, for the lift
93
p̃ of p near p̃0 , we have
|p̃0 − F̃ n (p̃)| >
p
l2 + δε2 .
Now, F n (γ) is a path joining F n (pε ) to F n (p) that stays within the ε neighborhood of
L+ (because the flow advances in the direction of L+ and γ ⊂ U0 ). The last inequality
implies that the ε neighborhood of F n (γ) actually contains the arc Ll . Since ε < ∆/2
and Ll was ∆/2-dense and ε + ε < ∆, F n (γ) and thus also U ⊃ F n (γ) are ∆-dense
in Td . By arbitrariness of ∆ > 0, cl(U) = Td .
2
Proof of Proposition 4:
As before, we consider the family of invariant measures µt := F∗t (µ), t ∈ R, and
denote their supports by Λt := supp(µt ). The plan is to show that, for any ǫ ∈ (0, 1),
the ǫ “thickening” of Λ0 ,
[
Λε :=
F t (Λ0 ),
t∈[0,ε]
contains an open and F -invariant subset. The lemma then will guarantee that Λε is
dense and so is Λ0 by arbitrariness of ǫ.
Let k be such that N ∩ [0, 1/ε + 1) = {0, . . . , k}. We map Λε by F nε and take a
union over 0 ≤ n ≤ k to get
ε
ε
ε
kε
ε
Λ ∪ F (Λ ) ∪ · · · ∪ F (Λ ) =
[
t
F Λ0 =
t∈[0,1]
[
supp(µt ) = supp
Z
0
t∈[0,1]
1
µt dt .
The last equality can be justified by the fact that for any open U ⊂ Td we have
Z
0
1
Z
µt dt (U) = 0 iff
1
µt (U) dt = 0 iff sup µt (U) = 0,
0
t∈[0,1]
94
where the integral was dropped based on the fact that if µt (U) = µ(F −t (U)) > 0
for some t then this it is so for all nearby t. (That in turn follows easily from the
regularity of the measure, openness of U, and continuity of F −t .)
Now,
R1
0
µt dt is the non-singular ergodic measure for the flow whose support is
all of Td . (Indeed, it has a positive density with respect to the Lebesgue measure on
Td , as follows from (3.5).) Thus, the union of the finitely many closed sets F nε (Λε ),
n = 0, . . . , k, is all of Td , forcing one of the sets to have a non-empty interior. In fact,
each F nε (Λε ) being an image of Λε under a homeomorphism, it must be that
U := int(Λε ) 6= ∅.
Moreover, F (U) = U because F (Λε ) = Λε (since already the Λt are F invariant). By
the lemma cl(U) = Td .
Finally, taking C > 0 to be an upper bound on the speed of the flow, the construction of Λε makes it clear that Λε is contained in the Cǫ-neighborhood of Λ0 ,
so
U ⊆ Λε ⊆ BCε (Λ0 ).
Therefore, we have
Td = cl(U) ⊆ cl(BCε (Λ0 )),
making Λ0 Cε-dense in Td . By arbitrariness of ǫ, Λ0 is dense in Td . That is supp(µ) =
Λ0 = Td (as it is closed).
2
95
Proof of Theorem 10:
As we explained, the theorem is an immediate corollary of the proposition. To
wit, the closure of the orbit of any point p ∈ Td is a closed invariant set, so has full
measure under some ergodic measure. By Proposition 4 the support of that measure
is either all of Td or the point p0 . If p 6∈ L− the support of the measure cannot be p0
so it must be all of Td . That is, the orbit of p ∈ Td \ L is dense in Td .
2
96
CHAPTER 4
FIXED POINT FREE MAPS
Bounded Deviation
For a SSM F without fixed points one may hope for the existence of a semiconjugacy to a rigid translation/rotation in a way analogous to Poincaré’s result for
circle homeomorphisms. In this chapter, we will show existence of the semi-conjugacy
for F that satisfy an additional hypothesis of bounded deviation: these F have a lift
F̃ with a unique average displacement vector ρ and the displacement of points under
iteration of F̃ differs from the displacement dictated by ρ by a bounded amount. The
precise definitions and our main theorem follow.
Definition 4 Given a map F̃ : D → D ⊆ Rm and vector ρ, we say that p̃ ∈ D has
bounded deviation with respect to F̃ if and only if there exists C > 0 such that for
any n ∈ Z
kF̃ n (p̃) − p̃ − nρk < C.
F̃ has bounded deviation if and only if every p̃ ∈ D has bounded deviation with respect
to F̃ .
Definition 5 A map F : Tm → Tm is said to have bounded deviation if there exists
a lift of F , F̃ : Rm → Rm such that F̃ has bounded deviation. In fact, if one lift of F
has bounded deviation then so do all other lifts.
97
Theorem 11 If an SSM, F : T2 → T2 , is a fixed point free homeomorphism with
bounded deviation, then F is semi-conjugate to an irrational translation.
As we mentioned in the introduction, this result also follows from [19] and is similar
to [46], works that were not at our disposal at the time of the development of our
Theorem 11.
We note that any F that is semi-conjugate to a rigid translation has bounded
deviation. On the other hand, the usefulness of the theorem is limited by the fact
that, given a concrete map F , it may be very hard to verify the bounded deviation
hypothesis.
For the duration of the chapter we assume that F is a map that satisfies the
hypothesis of the theorem. The rest of this chapter is devoted to the proof of the
theorem.
Let F̃ be the lift of F that fixes the leaves of the lifted foliation. By Theorem 4,
F̃ has a unique (non-zero) average displacement vector ρ = (ρθ , ρx ). By possibly conjugating by x̃ 7→ −x̃ or θ̃ 7→ −θ̃ we will assume that (ρθ , ρx ) has positive components.
F̃ is of the form
F̃ (θ̃, x̃) = (F̃x̃ (θ̃), F̃θ̃ (x̃)).
where F̃θ̃ (x̃) :=πx̃ ◦ F̃ (θ̃, x̃) and F̃x̃ (θ̃) :=πθ̃ ◦ F̃ (θ̃, x̃) and πθ̃ (θ̃, x̃) = θ̃ and πx̃ (θ̃, x̃) = x̃
are the projections. Moreover, since F̃ preserves the foliation in the direction α =
(ω, 1), where ω :=ρθ /ρx , if p̃ = (θ̃, x̃), we have
F̃θ̃ (x̃) = fp (x̃)
and
F̃x̃ (θ̃) = ω(fp (x̃) − x̃) + θ̃.
98
Above, as in Chapter 2, p = π(p̃) is the projection of p̃ onto T2 and fp : R → R is
the restriction of F to the foliation leaf through p and transferred to a map of R via
the parametrization of this leaf (see 2.2).
We note that x̃ 7→ F̃θ̃ (x̃) = fp (x̃) is an increasing homeomorphism1 and so is
θ̃ 7→ ω(fp (x̃) − x̃) + θ̃. In particular,
F̃ (θ̃, x̃) = (ω(fp (x̃) − x̃) + θ̃, fp (x̃))
preserves order within each line in the α direction.2 Adopting the following notations
for the components of the iterated map F̃ ,
F̃x̃n (θ̃) := πθ̃ ◦ F̃ n (θ̃, x̃)
and F̃θ̃n (x̃) := πx̃ ◦ F̃ n (θ̃, x̃),
and computing the average displacement for each component map yields the components of the rotation vector:
1 n
F̃x̃ (θ̃)
n→∞ n
ρθ = lim
and
1 n
F̃ (x̃),
n→∞ n θ̃
ρx = lim
where the limits are independent of x̃ and θ̃.
In general a map with bounded deviation will have component maps with bounded
n
n
deviation, by which we mean that F̃x̃ (θ̃) − θ̃ − nρθ and F̃θ̃ (x̃) − x̃ − nρx are uniformly bounded in n ∈ Z for all (θ̃, x̃). Conversely, a map in which all component
maps have bounded deviation will have bounded deviation itself. The geometries of
the orbits for an SSM allow the converse statement to be strengthened:
1
We also observe that fp (x̃) has positive displacement since non-zero rotation and Proposition 1
assures us that, for any x̃ ∈ R, fp (x̃) > x̃.
2
Of course we can also see this directly (without the breakdown to f˜p̃ ) by looking at the restriction
of the homeomorphism F̃ to a line [p̃] and noting that the restriction is a homeomorphism.
99
Fact 11 F̃ has bounded deviation if and only if F̃x̃ or F̃θ̃ has bounded deviation.
Proof of Fact 11:
It suffices to observe that, for any (θ̃, x̃), the identity ω(F̃θ̃ (x̃) − x̃) + θ̃ = F̃x̃ (θ̃)
ensures that all three deviations are multiples of each other:
and
n
F̃x̃ (θ̃) − θ̃ − nρθ = ω(F̃θ̃n (x̃) − x̃) + θ̃ − θ̃ − nρθ = ω F̃θ̃n (x̃) − x̃ − nρx n
F̃ (θ̃, x̃) − (θ̃, x̃) − n(ρθ , ρx ) = (F̃x̃n (θ̃) − θ̃ − nρθ , F̃θ̃n (x̃) − x̃ − nρx )
= F̃θ̃n (x̃) − x̃ − nρx |(ω, 1)| .
A look at Figure 6 makes the rest clear.
2
To prove the theorem, we will proceed by showing that F is flow-equivalent to
a ρ-bounded quasi-periodically forced map T . This means that T is constructed as
a certain first return map for the suspension flow of F . By Theorem 3, T will be
semi-conjugate to a rotation, which we will push back through the flow equivalence
to form a semi-conjugacy for the map F .
Flow Equivalence
We first create the flow (G̃t )t∈R on R3 (with coordinates (θ̃, x̃, ỹ)) that factors to
a flow on the torus R3 /Z3 which is conjugate to the suspension flow of F . The new
100
C
C2
F̃x̃n (θ̃)
θ̃ + nρθ
≤ C1
C1
≤C
≤ C2
θ̃
x̃ + nρx F̃θ̃n (x̃)
x̃
Figure 6. Picture proof of fact 11.
coordinate ỹ is such that, along any orbit under (G̃t ), it increases at the same rate as
t. As the orbit passes through each “integer floor” of R3 , given by
R2ỹ=k̃ := {(θ̃, x̃, ỹ) : ỹ = k̃},
(k̃ ∈ Z)
the trajectory is adjusted so that the holonomy between floors is given by (θ̃, x̃, k̃) 7→
(F̃ (θ̃, x̃), k̃ + 1).
Suspension Flow
To define G̃t formally we proceed in the following way. First, we describe how to
flow for a “short time” from an integer valued “floor”. So for 0 ≤ t ≤ 1,
G̃t (θ̃, x̃, ⌊ỹ⌋) := (θ̃ + t(F̃x̃ (θ̃) − θ̃), x̃ + t(F̃θ̃ (x̃) − x̃), ⌊ỹ⌋ + t).
(4.1)
The idea is that the line segments formed by considering the set {G̃t (θ̃, x̃, ⌊ỹ⌋)}t∈[0,1]
101
˜ ∈ R × R × Z assemble into the entire foliation for the flow.
for each distinct (θ̃, x̃, ⌊y⌋)
Although combining (4.1) with the flow property already determines G̃t , we will write
explicit formulas for G̃t for all t.
Flowing from an integer valued “floor” by a time t ∈ Z is to be equivalent to
applying G̃1 t-times so we set
G̃t (θ̃, x̃, ⌊ỹ⌋) := (F̃x̃t (θ̃), F̃θ̃t (x̃), ⌊ỹ⌋ + t).
If ỹ 6∈ Z we need to find the correct line segment to flow on. So we define
G̃−(ỹ−⌊ỹ⌋) (θ̃, x̃, ỹ) :=(θ̄, x̄, ⌊ỹ⌋)
where θ̄ and x̄ are the unique numbers such that
G̃ỹ−⌊ỹ⌋ (θ̄, x̄, ⌊ỹ⌋) = (θ̃, x̃, ỹ).
Existence of θ̄ and x̄ follows from continuity of F̃ and uniqueness is guaranteed because
the flow lines defined in (4.1) do not cross. This last point is a simple consequence
of the orientation preservation of F restricted to any individual leaf but we carefully
verify it below.
To see that the flow lines do not cross, first notice the projection (via πθ̃,x̃ :
(θ̃, x̃, ỹ) → (θ̃, x̃)) of each flow line onto the θ̃, x̃ coordinates equals the lift of a leaf of
the original foliation of F , henceforth called a leaf itself. We proceed by contradiction
and assume that flow lines do cross at some (θ̃, x̃, ỹ). We already know ỹ 6∈ Z by
construction (since F̃ is injective) so we may find two points (both on the same leaf
102
G̃t
ỹ
θ̃
F̃
x̃
Figure 7. Pictured are the first three iterates of a point on a ”floor” under the action
of F̃ . Also shown are the ”short” flow lines of G̃t (as defined in 4.1) from each iterate.
The orbit of the starting point is also indicated under G̃t for 0 ≤ t ≤ 3 .
103
that the projection of (θ̃, x̃, ỹ) is on), (θ̄1 , x̄1 , ⌊ỹ⌋) and (θ̄2 , x̄2 , ⌊ỹ⌋), where we have
Gỹ−⌊ỹ⌋ (θ̄1 , x̄1 , ⌊ỹ⌋) = Gỹ−⌊ỹ⌋ (θ̄1 , x̄1 , ⌊ỹ⌋) = (θ̃, x̃, ỹ). We can now use these points to
find the values
F̃θ̄1 (x̄1 ) = πx̃ ◦ G̃1 (θ̄1 , x̄1 , ⌊ỹ⌋) = x̄1 +
x̃ − x̄1
ỹ − ⌊ỹ⌋
F̃θ̄2 (x̄2 ) = πx̃ ◦ G̃1 (θ̄2 , x̄2 , ⌊ỹ⌋) = x̄2 +
x̃ − x̄2
.
ỹ − ⌊ỹ⌋
and
As part of the same leaf we have F̃θ̄1 = F̃θ̄2 = fp (where p̃ can be taken to be any
point of the common leaf). Observing that 1/(ỹ − ⌊ỹ⌋) > 1 allows us to see that the
difference
F̃θ̄1 (x̄1 ) − F̃θ̄2 (x̄2 ) = fp (x̄1 ) − fp (x̄2 ) = (x̄1 − x̄2 ) 1 −
1
ỹ − ⌊ỹ⌋
implies that fp̃ is not increasing, giving us our contradiction and verifying the uniqueness of θ̄ and x̄.
Finally, to finish defining G̃t for a general t ∈ R, we flow backwards to the nearest
integer-valued “floor”, move by integer leaps to a new “floor” then flow the remaining
short distance along the linear piece associated with the final floor. Precisely,
G̃t (θ̃, x̃, ỹ) := G̃s−⌊s⌋ ◦ G̃⌊s⌋ ◦ G̃−(ỹ−⌊ỹ⌋) (θ̃, x̃, ỹ) with s := t + ỹ − ⌊ỹ⌋.
Establishing T
From the construction, it is clear that the suspension flow of F̃ is conjugate to the
factor of G̃t by the group of the integer translations in the direction of the ỹ-axis. To
104
F̃θ̃ (x̄2 )
F̃θ̃ (x̄1 )
(θ̃, x̃, ỹ)
x̄1
x̄2
Figure 8. A depiction of how F̃θ̃ ceases to be increasing if flow lines were to cross.
perform the flow equivalence and generate a suitable skew product map, we consider
T̄ obtained as the return map along the flow to the family of planes with integer x̃
coordinate,
R2x̃=k̃ := {(θ̃, x̃, ỹ) : x̃ = k̃},
(k̃ ∈ Z).
The map T̄ is defined by sending a point in the plane R2x̃=k̃ to the first hit of its flow
trajectory into the next plane. Such a “first hit” exists because F is fixed point free
and ρx > 0. This ensures that the x̃-coordinate of orbits of (G̃t ) increases with some
positive minimal speed and tends to infinity,
lim πx̃ ◦ G̃t (θ̃, x̃, ỹ) = ∞.
t→∞
105
The map T̄ is of the form
T̄ : (θ̃, k̃, ỹ) 7→ (θ̃ + ω, k̃ + 1, T̃θ̃ (ỹ))
where T̃θ̃ : R → R. One can see that T̄ is continuous with a continuous inverse and
thus a homeomorphism. This hinges on the fact that the flow lines intersect the
planes transversally (due to “the positive minimal speed”).
Because (G̃t ) is Z3 -equivariant, by taking all coordinates modulo one and dropping
the middle coordinate (since it is constant 0 mod 1), the map T̄ induces a map
T : T2 → T2 which is of the form
T : (θ, y) 7→ (θ + ω, Tθ (y)).
Figure 9 depicts the situation including the lift T̃ of T that is a factor of T̄ , via
πθ,y : (θ̃, x̃, ỹ) → (θ, y) = π(θ̃, ỹ). T is a skew-product of circle maps Tθ . As is usual
we will use T̃θ̃n (ỹ) to denote πỹ ◦ T̃ n (θ̃, ỹ).
Before showing that T is ρ-bounded we first record a relationship between the
rotation vectors of F̃ and T̃ . Let ρT̃ be the ỹ-component of the average displacement
vector of T̃ so that
1 n
(T̃ (ỹ) − ỹ) → ρT̃ ,
n θ̃
(θ̃, ỹ) ∈ R2 .
Fact 12 We have
ρT̃ = 1/ρx .
Proof of Fact 12:
We define r̃(θ̃, ỹ) as the travel time from one wall to the next, that is, for
106
ω
T̃
ρT̃
1
ỹ
θ̃
ρθ
ρx
F̃
x̃
Figure 9. How T̃ is constructed via F̃ and how the various average displacements
interplay. Note that the elevation labeled ρT̃ is purposely “mislabeled” as the average
tn
→ ρT̃ .
ỹ displacement, the distance shown is tn=1 , as n grows we have
n
107
(θ̃, ỹ) ∈ R2 ,
G̃r̃(θ̃,ỹ) (θ̃, 0, ỹ) = T̄ (θ̃, 0, ỹ) = (θ̃ + ω, 1, T̃θ̃ (ỹ)).
Then the time to the n-th wall is
tn (θ̃, ỹ) = tn :=
n−1
X
r̃(θ̃ + iω, T̃θ̃i (ỹ)),
i=0
and we have that
G̃tn (θ̃, 0, ỹ) = (θ̃ + nω, n, T̃θ̃n (ỹ)).
By construction, ỹ increases at the same rate as t, so
tn = T̃θ̃n (ỹ) − ỹ.
(4.2)
We also have that
ρx = lim
t→∞
1 t
1
n
F̃θ̃ (0) = lim
πx̃ (G̃tn (θ̃, 0, ỹ) − (θ̃, 0, ỹ)) = lim .
t
→∞
n→∞
n
t
tn
tn
Thus we have that tn /n → ρT̃ and n/tn → ρx and the fact follows.
2
Since
ρT̃ = 1/ρx
and
ω = ρθ /ρx ,
(4.3)
it is easy to see that AρT̃ + Bω = C if and only if A + Bρθ = Cρx . Thus, ρT̃ and ω
are rationally independent if and only if ρx and ρθ are. Of course, as F is a SSM, ρx
and ρθ are rationally independent. Hence, in order to apply Theorem 3 to guarantee
that T is semi-conjugate to an irrational torus-translation, all we have left to verify
is that T is ρ-bounded.
108
ρ-Bounded T
Observing,
n
n
T̃
(
θ̃,
x̃)
−
(
θ̃,
x̃)
−
n(ω,
ρ
)
=
θ̃
+
nω
−
θ̃
−
nω,
T̃
(x̃)
−
x̃
−
nρ
T̃ T̃ θ̃
allows us to state:
n
= T̃θ̃ (x̃) − x̃ − nρT̃ ,
Fact 13 A quasiperiodically forced circle map T is ρ-bounded if and only if T has
bounded deviation. Furthermore, the constant C bounding the deviation is the same.
So in order to show that T is ρ-bounded we may proceed by showing that it has
bounded deviation, which the following lemma provides.
Lemma 15 F has bounded deviation if and only if T has bounded deviation. Furthermore the bounding constant C may be chosen to be independent of the choice of
p̃ = (θ̃, x̃).
This lemma will be a direct consequence of (multiple applications of) lemma 16,
which will be stated shortly in terms of a new function Ic . This new function is
defined so that with one choice of the parameter c it is a lift of F and with a different
c, a lift of T .
Let c be any one of the three symbols θ̃, x̃, ỹ. Picking c is simply picking one of
the three standard coordinates on R3 . (So c is a string, c ∈ {“θ̃”, “x̃”, “ỹ”} and not
109
c ∈ {θ̃, x̃, ỹ}.) We define a family of planes
Fc :=
[
R2c=k̃ ,
k̃∈Z
where,
R2c=k̃ := {(θ̃, x̃, ỹ) : c = k̃ ∈ Z}.
Fix a point p̃′ ∈ R3 and let Q be the orbit of that point under the flow G̃t . Define
Ic : Q
T
Fc 7→ Q
T
Fc so that for any p̃ ∈ Q
T
Fc we have πc ◦Ic (p̃) = πc (p̃)+πc (1, 1, 1).
The orbit Q being arbitrary, this defines Ic on all of Fc and Ic is the first return map to
Fc under G̃t , just like T̄ . In fact, Ix̃ = T̄ and one can argue that Ic is a homeomorphism
just like for T̄ . The average displacement vector for Ic is in the same direction as that
of G̃t and since πc (Ic (p̃) − p̃) = 1, πc (Icn (p̃) − p̃) = n is equal to
Ick (p̃) − p̃
(ρθ , ρx , 1)
=
k→∞
k
πc (ρθ , ρx , 1)
ρIc := lim
where p̃ is any point in Fc .
Lemma 16 Let p̃′ ∈ R3 be fixed. Denote by Q the flow orbit of p̃′ (as before) and
by L the line L :={p̃′ + t(ρθ , ρx , 1) : t ∈ R}. Every point p̃ ∈ Q
T
Fc has bounded
deviation with respect to Ic if and only if there is a CQ > 0 such that for any q̃ ∈ Q
there exists ˜l ∈ L with
|q̃ − ˜l| ≤ CQ .
That is, every point of Q is no further than CQ from a point in L.
Proof of Lemma 16:
This is pretty clear but we will again give a detailed argument so we may
110
track the interplay between the bounding constants.
We first show (⇐=). Since
Icn (p̃) ∈ Q we may find a ˜l ∈ L where |Icn (p̃) − ˜l| ≤ CQ . We also find ˜ln ∈ L such
that πc (˜ln ) = πc ◦ Icn (p̃). Then letting b be the unit basis vector in the coordinate c
we have from (right) triangle geometry (see Figure 10) that
|˜l − ˜ln | · πc (ρθ , ρx , 1)
p
CQ ≥ |Icn (p̃) − ˜l| ≥ |˜l − (˜l − bπc (˜l − ˜ln ))| =
.
1 + ρ2θ + ρ2x
In particular, we have
|˜l − ˜ln | ≤ CQ
p
1 + ρ2θ + ρ2x
=: Cc .
πc (ρθ , ρx , 1)
(4.4)
We note that Cc is dependent upon3 c and our choice of Q (which dictates CQ ), both
of these are fixed in the context of the lemma. Since the lines L and {p̃ + t(ρθ , ρx , 1) :
t ∈ R} have the same direction vector the distance between them (dependent only on
p̃) is fixed. In any of the R2c=k̃ planes we find the distance between the two lines and
denote it by Cp̃ . (We note that in the special case, p̃′ = p̃, we have Cp̃ = 0.) Thus for
any p̃ ∈ Q
T
Fc setting
CI = CQ + Cc + Cp̃ ,
(4.5)
allows us to state that for any n
|Icn (p̃) − p̃ − nρIc | ≤ |Icn (p̃) − ˜l| + |˜l − ˜ln | + |˜ln − (p̃ + nρIc )| ≤ CI
and Ic has bounded deviation.
3
We could maximize over our three values of c, to remove that dependency, but we choose not to
do so.
111
G̃t (p̃′ ) = Q
L
πc (ρθ , ρx , 1)
˜l
Icn (p̃)
πd (ρθ , ρx , 1)
˜l − bπc (˜l − ˜ln )
˜ln
πe (ρθ , ρx , 1)
Figure 10. Here c, d, e ∈ {θ̃, x̃, ỹ} are each distinct coordinates of R3 . The (piece of)
plane depicted is a span of d and e, with Icn (p̃), ˜ln and ˜l − bπc (˜l − ˜ln ) elements of that
plane. The distance between Icn (p̃) and ˜l is of course bounded (above) by CQ .
112
G̃t (p̃′ ) = Q
G̃t (p̃′ ) = Q
L
R2c=k̃+1
R2c=k̃+1
p̃ + nρIc
Cp̃
R2c=k̃
CQ
L
C Gc
˜ln
Cc
˜l
R2c=k̃
q̃
Cp̃
˜ln
CI
˜l
Icn (p̃)
Icn (p̃)
p̃
p̃
Cp̃
Cp̃
Figure 11. Dashed lines indicate the lengths between q and l on the left and between
Icn (p̃) and p̃ + n(ρθ , ρx , 1) on the right. Each length is labeled not by the length itself
but by the bound on that length. Note that we have ln and p̃ + nρIc with the same
c-coordinate since πc (Icn (p̃) − p̃) = n = nπc (ρIc ).
To show (=⇒) we first fix any p̃ ∈ Q
such that for any n we have
T
Fc . By our hypothesis we may find CI
|Icn (p̃) − p̃ − nρIc | ≤ CI .
As before, when measured in any of the R2c=k̃ planes, we denote the distance between
L and {p̃ + t(ρθ , ρx , 1) : t ∈ R} by Cp̃ .
We will need to see that the distance along G̃t between R2c=k̃ and R2c=k̃+1 is
bounded. To this end we recall that the orbits of G̃t are made up of short line
segments the shortest of which is no shorter than (ωm, m, 1) and the longest at most
113
(ωM, M, 1). Where
m := min{f˜p (x̃) − x̃} and M := max{f˜p (x̃) − x̃}
(θ̃,x̃)
(θ̃,x̃)
with the minimums and maximums taken over all p̃ = (θ̃, x̃) ∈ [0, 1]2 . (Note that
m 6= 0 since f˜p is a homeomorphism without fixed points.) Between R2c=k̃ and R2c=k̃+1
there are at most s :=1 + ⌊1/πc (ωm, m, 1)⌋ segments so the distance between R2c=k̃
and R2c=k̃+1 on G̃t is less than
CGc :=ks(ωM, M, 1)k.
Thus, for any p̃ ∈ Q
T
Fc , we may set
CQ = CGc + CI + Cp̃ .
(4.6)
Then, for any q̃ ∈ Q, we find ˜l ∈ L with πc (˜l) = ⌊πc q̃⌋ and n ∈ Z with πc Icn (p̃) = ⌊πc q̃⌋.
So we have,
|q̃ − ˜l| ≤ |q̃ − Icn (p̃)| + |Icn (p̃) − (p̃ + nρIc )| + |(p̃ + nρIc ) − ˜l| ≤ CQ
and every point of Q is no further than CQ from a point in L.
2
Before proceeding we make note of the fact that
πθ,x ◦ Iỹ (θ̃, x̃, ỹ) = F (θ, x) and πθ,y ◦ Ix̃ (θ̃, x̃, ỹ) = T (θ, y).
Therefore, (similar to fact 13)
n
Iỹ (θ̃, x̃, ỹ) − (θ̃, x̃, ỹ) − n(ρθ , ρx , 1) = F̃ n (θ̃, x̃) − (θ̃, x̃) − n(ρθ , ρx )
114
and
n
Ix̃ (θ̃, x̃, ỹ) − (θ̃, x̃, ỹ) − n(ω, 1, ρT̃ ) = T̃ n (θ̃, ỹ) − (θ̃, ỹ) − n(ω, ρT̃ ) .
Thus, F has bounded deviation if and only if the orbits of Iỹ have bounded deviation
and for each orbit the bounds are the same. Similarly for T and Ix̃ .
We now prove Lemma 15 with 4 applications of Lemma 16.
Proof of Lemma 15:
We show (=⇒) explicitly; that is, that F with bounded deviation implies T
has bounded deviation. We select any (θ, y) ∈ T2 , lift it to p̃′ = (θ̃, 0, ỹ) ∈ [0, 1)3
and create Q and L as per the lemma. Since F has bounded deviation, for any
p̃ = (θ̃, x̃, ỹ) ∈ Q
T
Fy , we may find a constant CF (a priori dependent upon our
choice of θ̃ and x̃) so that for any n ∈ Z
n
n
Iỹ (p̃) − p̃ − n(ρθ , ρx , 1) = F̃ (θ̃, x̃) − (θ̃, x̃) − n(ρθ , ρx ) ≤ CF .
Thus we may apply Lemma 16 to Iỹ and find that for any q̃ ∈ Q there exists a ˜l ∈ L
such that |q̃ − ˜l| ≤ CQ = CGỹ + Cp̃ + CF .
Using the same p̃′ (hence the same Q and L) as before we apply Lemma 16 again,
this time to Ix̃ . From the previous paragraph we know for any q̃ ∈ Q there exists a
˜l ∈ L where |q̃ − ˜l| ≤ CQ . So for p̃′ = (θ̃, 0, ỹ) ∈ Q T Fx̃ ,
n
T̃ (θ̃, ỹ) − (θ̃, ỹ) − n(ω, ρT̃ ) = kIx̃n (p̃′ ) − (p̃′ ) − n(ω, 1, ρT̃ )k ≤ CT ,
115
where,
p
1 + ρ2θ + ρ2x
′
CT = Cx̃ + Cp̃ + CQ = CQ
+ CQ
πx̃ (ρθ , ρx , 1)
!
p
1 + ρ2θ + ρ2x
= CGỹ + Cp̃ + CF
1+
πx̃ (ρθ , ρx , 1)
(4.7)
(For the first equality we note we are in the special case Cp̃′ = 0.) Since (θ, y) ∈ T2
was arbitrary, the orbit of the lift of any such point has bounded deviation under Ix̃ ,
and we have that T has bounded deviation.
Interchanging the rolls of x̃, ỹ and F, T a similar argument shows (⇐=) that T
with bounded deviation implies that F has bounded deviation. In particular we may
rewrite (4.7) into
CF = (CGx̃
p
1 + ρ2θ + ρ2x
+ Cp̃ + CT ) 1 +
πỹ (ρθ , ρx , 1)
!
However, Theorem 2 allows us to find a uniform bound on the deviation for T . That
is, we may choose the constant CT such that
n
T̃ (θ̃, ỹ) − (θ̃, ỹ) − n(ω, ρT̃ ) ≤ CT
for all (θ̃, ỹ) ∈ R2 and n ∈ Z.
Additionally, we note that the value of Cp̃ in the proof of the lemma was dependent upon a particular (fixed) choice of p̃ ∈ Q
T
Fx̃ . Choosing p̃ so that πx̃ (p̃) = 0
guarantees that p̃ and p̃′ are no further than CGx̃ apart. Constructing a triangle
between the points p̃′ , p̃ and (p̃′ ) − (x̃/ρx̃ )(ρθ̃ , ρx̃ , 1) we see that
Cp̃ ≤ CGx̃ + k
x̃
(ρ , ρx̃ , 1)k ≤ CGx̃ + k(ω, 1, ρT̃ )k.
ρx̃ θ̃
116
Therefore, we find that our bound, on the deviation of F ,
q
CF = (CGx̃ + Cp̃ + CT ) 1 + 1 + ρ2θ + ρ2x
q
2
2
≤ (CGx̃ + CGx̃ + k(ω, 1, ρT̃ )k + CT ) 1 + 1 + ρθ + ρx
is independent of our choice of (θ, x) so CF is the uniform bound on the deviation F
that we desired.
2
Semi-Conjugacy
In the lift to R3 what we have done, in essence, is taken our SSM with bounded
deviation on the “floors” and induced a ρ-bounded skew product on the “walls”. We
may now use Theorem 3 to get a semi-conjugacy on the “walls”, use that to construct
a semi-orbit equivalence4 between (G̃t ) and the linear flow on the whole of R3 , and
finally extract it to the “floors”, thereby generating the sought after semi-conjugacy
of F to the rotation.
Conclusion of Proof of Theorem 11:
Since F has bounded deviation T is ρ-bounded by Lemma 15 and Fact 13. We
denote by h the semi-conjugacy, from Theorem 3, between T and Rω,ρT̃ .
This h
preserves the fibers of the skew product. We let h̃ : R2 → R2 be a lift of h (so that
π(h̃(θ̃, ỹ)) = h(π(θ̃, ỹ))). The semiconjugacy property, h ◦ T = Rω,ρT̃ ◦ h, translates
into h̃ ◦ T̃ = Rω,ρT̃ ◦ h̃ where we overloaded the notation letting Rω,ρT̃ stand for the
4
That is, a semi-conjugacy up to a time change
117
translation on R2 given by
Rω,ρT̃ (θ̃, ỹ) = (θ̃ + ω, ỹ + ρT̃ ).
(Note that, in general, h̃ ◦ T̃ and Rω,ρT̃ ◦ h̃ could differ by a deck translation but this
is not the case because T̃ and Rω,ρT̃ have the same rotation vector.)
Since Gt is a flow for all (θ̃, x̃, ỹ) ∈ R3 we may find unique θ̃0 , ỹ0 and tx̃ where
(θ̃0 , ⌊x̃⌋, ỹ0) = G̃−tx̃ (θ̃, x̃, ỹ). We use these to extend h̃ to H̄ on R3 as follows. We
define
tx̃
r(θ̃0 ,ỹ0 )
H̄(θ̃, x̃, ỹ) := R⌊x̃⌋
◦ h̃(θ̃0 , ỹ0 )
(4.8)
where
Rx̃t (θ̃, ỹ) = (θ̃ + tω, x̃ + t, ỹ + tρT̃ ).
In essence, we see where we are and how long it takes to flow backwards to the
“wall” R2x̃=⌊x̃⌋ , we use h̃ to move from there to a new spot on the “wall”, then flow
forward in the (ω, 1, ρT̃ ) direction using the linear flow
Rt (θ̃, x̃, ỹ) :=Rx̃t (θ̃, ỹ)
for an amount of time scaled to account for the fact that the time of flight between
walls R2x̃=⌊x̃⌋ and R2x̃=⌊x̃⌋+1 is r(θ̃0 , ỹ0) for the flow Gt and constant equal to 1 for the
flow Rt .
To see that H̄ is continuous we first note, from (4.2), that r̃ is the composition of
continuous functions
r̃(θ̃, ỹ) = t1 = T̃θ̃ (ỹ) − ỹ
118
and is thus continuous itself. Furthermore, when we restrict the x̃ coordinate to stay
in any one fundamental domain of the form [0, 1)+Z then θ̃0 , ỹ0 and tx̃ depend continuously on θ̃, x̃, ỹ, so on those restricted domains the continuity of H̄ is a consequence
of the continuity of its pieces.
To get continuity on the entire domain we need to examine the behavior of H̄ on
consecutive ”walls”, R2x̃=k̃ and R2x̃=k̃+1 . For k̃ ∈ Z and θ̃, ỹ ∈ R, we have
R1 ◦ H̄(θ̃, k̃, ỹ) = R1 ◦ Rk̃0 ◦ h̃(θ̃, ỹ)
0
= Rk̃+1
◦ Rω,ρT̃ ◦ h̃(θ̃, ỹ)
0
= Rk̃+1
◦ h̃ ◦ T̃ (θ̃, x̃)
= H̄(θ̃ + ω, k̃ + 1, T̃θ̃ (ỹ))
= H̄ ◦ T̄ (θ̃, k̃, ỹ).
(4.9)
The “join up” between the fundamental dominans, upon which we already know H̄
is continuous, can be seen to be continuous by examining
kH̄(θ̃, k̃, ỹ) − H̄(θ̃, k̃ − ε, ỹ)k
for k̃ ∈ Z and θ̃, ỹ ∈ R and 0 < ε ≪ 1. From the definitions of θ̃0 , ỹ0 and T̄
(θ̃, k̃, ỹ) = G1 (θ̃0 , k̃ − 1, ỹ0 ) = T̄ (θ̃0 , k̃ − 1, ỹ0).
(4.10)
119
So using that and (4.9) we have
t
k̃−ε
r(θ̃0 ,ỹ0 )
kH̄(θ̃, k̃, ỹ) − H̄(θ̃, k̃ − ε, ỹ)k = kH̄ ◦ T̄ (θ̃0 , k̃ − 1, ỹ0 ) − R⌊k̃−1⌋ ◦ h̃(θ̃0 , ỹ0 )k
t
k̃−ε
r(θ̃0 ,ỹ0 )
1
= kR ◦ H̄(θ̃0 , k̃ − 1, ỹ0 ) − R⌊k̃−1⌋ ◦ h̃(θ̃0 , ỹ0 )k
=
Since tk̃−ε → r(θ̃0 , ỹ0 ) as ε → 0,
1
kRk̃−1
tk̃−ε
r(θ̃0 ,ỹ0 )
t
k̃−ε
r(θ̃0 ,ỹ0 )
◦ h̃(θ̃0 , ỹ0 ) − R⌊k̃−1⌋ ◦ h̃(θ̃0 , ỹ0 )k
→ 1. So (4.10) approaches zero as ε → 0
and continuity of H̄ is established. This continuity as well as (4.9) establishes that H̄
restricted to the walls is a semi-conjugacy between T̄ and the rotation R1 . Thus H̄
maps the orbits of the flow (G̃t ) to the orbits of the translation flow (Rt ). (By the very
construction of H̄, we knew that it mapped the pieces of an orbit of (G̃t ) between the
walls to the corresponding pieces for (Rt ), but it takes this semi-conjugacy property
to assure that the pieces assemble properly into the whole orbits.)
We note that G̃t and Rt need not be conjugated by H̄ since the “time-scaling” is
only corrected for when both image and preimage under the flow have integer values
of x̃. However, all we need is that the flow-lines map under H̄ to flow-lines. This
allows us to define a “time change” function α : R3 × R → R such that
H̄ ◦ G̃t (θ̃, x̃, ỹ) = Rα(θ̃,x̃,ỹ,t) ◦ H̄(θ̃, x̃, ỹ).
We now define the continuous function Ĥ : (θ̃, x̃, ỹ) 7→ R−t∗ H̄(θ̃, x̃, ỹ) where t∗ is
such that for any (θ̃, x̃, ỹ) ∈ R3 ,
πỹ ◦ Ĥ(θ̃, x̃, ỹ) = ỹ.
120
Continuity comes from the continuity of the elements we compose to make Ĥ after
we verify that t∗ exists and depends continuously on its arguments. The latter we
can see from
ỹ = πỹ Ĥ(θ̃, x̃, ỹ) = πỹ H̄(θ̃, x̃, ỹ) − t∗ ρT̃ ,
leading to
t∗ =
1 πỹ H̄(θ̃, x̃, ỹ) − ỹ .
ρT̃
This last is again a composition of continuous functions.
As with H̄, the function Ĥ maps flow lines of G̃t to flow lines of Rt . But Ĥ has
the important extra characteristic of preserving the ỹ coordinate. So the two points
p̃ and G̃1 (p̃) (trivially on the same G̃t flow-line) with ỹ-coordinates one unit apart
have images under Ĥ that also have ỹ-coordinates one unit apart. These images are
on the flow-line Rt (in the direction (ω, 1, ρT̃ )), so we have that the differences in the
θ̃ and x̃-coordinates of these two points are ρθ and ρx respectively. Thus,
Rρθ ,ρx ,1 ◦ Ĥ(θ̃, x̃, ỹ) = Ĥ ◦ G̃1 (θ̃, x̃, ỹ).
(4.11)
One more characteristic we need, that Ĥ has (see Figure 12), is
Ĥ(θ̃, x̃, ỹ) = Ĥ(θ̃, x̃, ỹ + 1) − (0, 0, 1).
(4.12)
Indeed, Ĥ is equivariant under any Z3 translations, which follows from the Z3 equivariance of everything in its construction. In particular, (to verify 4.12) the
equivariance of Gt is used when tx̃ = ỹ − ỹ0 to see
G̃−tx̃ (θ̃, x̃, ỹ) = (θ̃0 , ⌊x̃⌋, ỹ0 ) = G̃−tx̃ (θ̃, x̃, ỹ + 1) − (0, 0, 1).
121
(πθ̃ ĥ(θ̃0 , ỹ0 + 1), ⌊x̃⌋, πỹ ĥ(θ̃0 , ỹ0 + 1))
(θ̃, x̃, ỹ + 1) 7→ Ĥ(θ̃, x̃, ỹ + 1)
(θ̃0 , ⌊x̃⌋, ỹ0 + 1)
(θ̃, x̃, ỹ) 7→ Ĥ(θ̃, x̃, ỹ)
(θ̃0 , ⌊x̃⌋, ỹ0 )
(πθ̃ ĥ(θ̃0 , ỹ0), ⌊x̃⌋, πỹ ĥ(θ̃0 , ỹ0 ))
(θ̃, x̃, ỹ) 7→ (F̃ (θ̃, x̃), ỹ)
Figure 12. To see the points (θ̃, x̃, ỹ) and (θ̃, x̃, ỹ + 1) map under Ĥ to images one
ỹ-unit apart (4.12), we may trace their orbit under G̃t back to the ”wall” ⌊x̃⌋, the
crossing points are dictated by the orbit of the point under F̃ shown at the base.
Both images under ĥ have the same θ̃ and x̃-coordinates and are one unit apart as
are those points after flowing by Rt to the correct ỹ-value.
Using (4.11) and (4.12) will allow us to see that defining
H(θ, x) :=πθ,x ◦ Ĥ(θ̃, x̃, 0)
(4.13)
122
is the desired function so that Rρθ ,ρx ◦ H = H ◦ F . Indeed
Rρθ ,ρx ◦ H(θ, x) = Rρθ ,ρx ◦ πθ,x ◦ Ĥ(θ̃, x̃, 0)
= Rρθ ,ρx ◦ πθ,x ◦ (Ĥ(θ̃, x̃, −1) − (0, 0, −1))
= πθ,x ◦ Rρθ ,ρx ,1 ◦ Ĥ(θ̃, x̃, −1)
= πθ,x ◦ Ĥ ◦ G̃1 (θ̃, x̃, −1)
= πθ,x ◦ Ĥ ◦ (F̃ (θ̃, x̃), 0)
= H ◦ F (θ, x).
So we have that F is semi-conjugate with the rigid rotation Rρθ ,ρx .
2
123
APPENDICES
124
APPENDIX A
CONTINUED FRACTIONS
125
The regular (simple) continued fraction expansion for an irrational number β > 0
is given by
1
β = a0 +
,
1
a1 +
a2 +
1
..
.
were (ai ) is a sequence of non-negative integers, (ai ) ⊂ Z+ . We call (ai ) the coefficients (for the continued fraction expansion) of β. We shorten the typesetting by
the notation β := cf [a0 ; a1 a2 a3 . . . ] or even shorter β := cf [ai ]. Given any sequence of
non-negative integers the convergence of cf [ai ] is shown in [43] or other standard texts
on continued fractions. We note that when a0 = 0 and a1 6= 0 we have β ∈ [0, 1].
Since this is the range of numbers we will be working with we will drop the starting
0 and refer to
cf [a0 ; a1 a2
. . . ] with a0 = 0 as
cf [a1 a2
. . . ].
Rational approximations to β are given by the convergents of the continued fraction defined as
1
ci = cf [a1 a2 . . . ai ] =:
.
1
a1 +
a2 +
1
..
.+
1
ai
If we express the ci as reduced fractions,
ci = pi /qi ,
GCD(pi , qi ) = 1,
then the numerators pi ∈ Z+ and the denominators qi ∈ N can be found recursively
from
pi+1 = ai+1 pi + pi−1
and
qi+1 = ai+1 qi + qi−1
(i ∈ Z+ )
126
seeded with the initial conditions p−1 = 1, p0 = 0, q−1 = 0 and q0 = 1.
The recursive nature of qi is enough to give us the following bound on the minimum growth of the denominators of our convergents. The statement and proof are
taken from [43].
Fact 14 If qi are the denominators of the convergents of some number β, then for
i≥2
qi ≥ 2
i−1
2
.
Proof of Fact 14:
We know
qi = ai qi−1 + qi−2 ≥ qi−1 + qi−2 ≥ 2qi−2 .
After (i − 1)/2 successive applications we have one of two cases:
qi ≥ 2
i−1
2
q0 ,
when i is even or
qi ≥ 2
i−1
2
q1 ,
for i odd. Since q1 > q0 = 1, in either case we have
qi ≥ 2
i−1
2
.
2
127
Right Interval
Left Interval
Right Interval
Left Interval
β
β−1
0
2β − 1
β
0
2β − 1
β
2β − 1
β−1
0
β
β−1
β−1
Figure 13. Two depictions of the Rβ rotation indicating the left and right intervals.
Finding Convergents
Continued fraction expansions have an intimate connection with rotation by β of
the circle T = R/Z. Adopting [β − 1, β) as the fundamental domain, the rotation by
β is given by
Rβ (x) =
x+β
for x ∈ [β − 1, 0),
x+β−1
for x ∈ [0, β).
For x ∈ R we also define
kxk := min{|x − z| | z ∈ Z)}.
If x ∈ [β − 1, 0) we will sometimes say that x is in the left interval, and similarly if
x ∈ [0, β) we say that x is in the right interval.
To fix attention it is convenient to assume that the left interval is shorter than
128
the right which amounts to requiring that β ∈ (.5, 1). This is to say that a1 = 1. The
case β ∈ (0, .5) is redundant in the sense that it can be treated by using the rotation
by 1 − β∈ (.5, 1).
We will show how to recursively define the ai and qi starting from the initial
values a1 = 1, q0 = q1 = 1. Let
xn = Rβn (0),
(n ∈ Z+ ).
So x1 = xq1 = Rβ (x0 ). In the first step, we define a2 ∈ Z+ as the minimal value such
that
kRβa2 (xq1 )k < kxq1 k
and set
q2 = a2 + 1 = a2 q1 + q0 .
Clearly,
kxq2 k < kxq1 k.
To continue, for i ≥ 3, we proceed by induction. We assume that a1 , . . . , ai−1 ,
q1 , . . . , qi−1 are already defined and take ai ∈ Z+ to be minimal value such that
a qi−1
kRβi
(xqi−2 )k < kxqi−1 k.
and set
qi = ai qi−1 + qi−2 .
Again,
kxqi k < kxqi−1 k.
129
Rβq1
Rβq1
x0
xq1
Rβq1
Rβq2
xq1
Rβq2
Rβq1
xq2
x0
Rβq3
xq1
xq3
xq1
xq3
x0
Rβq4
Rβq3
x0 xq4
Rβq3
xq2
xq2
Figure 14. A depiction of the steps to find x4 and β =cf [1212 . . . ], that is the first
few coefficients of the continued fraction representation of β. We note that there are
two ways to ”reach” xqi shown at each step, each from a different direction. This
depicts the equivalence of ai qi−1 + qi−2 applications of Rβ1 to x0 = 0 and applying
q
Rβi−1 , ai -times to xqi−2 .
Claim 2 For i ∈ N, exactly one of xqi and xqi+1 is in the right interval. In fact, xqi
is in the right interval if and only if i is even.
Proof of 2:
We consider the action of Rβqi . When xqi is in the left interval Rβqi moves elements
a distance of kxqi k to the left. Similarly when xqi is in the right interval Rβqi moves
elements a distance of kxqi k to the right.
Initially, xq1 = β − 1 is the left endpoint of the left interval and jumps under
the action of Rβq1 = Rβ to the point Rβq1 (xq1 ) = 2β − 1, which is 1 − β to the
left of the right endpoint β. (Keep in mind that 1 − β < 0.5 < β.) Subsequent
130
application of Rβq1 (a2 − 1 times) will move the point to the left by 1 − β until no
more left translation by 1 − β within the right segment is possible because we arrived
at the point xq2 = Rβa2 q1 (xq1 ) which is closer to 0 than 1 − β, that is the condition
kxq2 k < kxq1 k is satisfied. Clearly, xq2 is in the right interval.
Having dealt with xq1 and xq2 we continue by induction. For i ≥ 2, we assume
a
q
that xqi−1 and xqi are in different intervals and look at how xqi+1 = Rβi+1 i (xqi−1 ) is
generated by repeated application of Rβqi . If xqi−1 is in the left(right) interval, then
the action of the rotation Rβqi moves this point to the right(left) translating by the
distance kxqi k. Since kxqi k < kxqi−1 k the first translation Rβqi (xqi−1 ) is still in the
left(right) interval, and this continues under subsequent (ai+1 − 1) applications of Rβqi
a
q
until we reach xqi+1 = Rβi+1 i (xqi−1 ) for which kxqi+1 k < kxqi k. Clearly, xqi−1 and xqi+1
are both in the same left(right) interval, while xqi was in the right(left) interval. This
completes the induction step.
2
131
APPENDIX B
OSTROWSKI’S β-NUMERATION
132
Given a continued fraction expansion (as found in Appendix A) for a fixed β ∈
(.5, 1)\Q, we shall describe how to uniquely encode the natural numbers in a way
that is intimately connected to the rotation by β and the small denominators. This
β-numeration is originally due to Ostrowski [44, 45] and is a normal U-representation
[47]. The basic idea of the expansion is to abandon a fixed base, that gives us the
exponential progression of orders of magnitude (like 1, 10, 100, . . .), and to use the
basis q1 , q2 , q3 ,. . . instead.
Fix m ∈ Z+ to be expanded. We set
m0 := m
and define a finite sequence m1 , m2 , . . . , mj recursively. Assuming that m0 , . . . , mi
are already determined, we set
Ni = max{k : qk ≤ mi }.
We then pick dNi ∈ {1, 2, . . . , aNi +1 } so that
0 ≤ mi − dNi qNi < qNi
and set
mi+1 := mi − dNi qNi .
One could say that we took the largest qk that can be bitten away from mi and
“greedily” took as many bites as possible.
133
This process is to be repeated until mj+1 = mj − dNj qNj = 0 (which is bound
to happen because the mi are decreasing and q1 = 1). Note that we devoured all of
m = m0 since
m0 = m1 + dN0 qN0 = m2 + dN1 qN1 + dN0 qN0 = . . . = dNj qNj + · · · + dN0 qN0 .
To smoothen the notation, we set dk = 0 for all the k ∈ Z+ \ {N1 , . . . , Nj }, call
the sequence (dk )∞
k=1 the β-expansion, and express the equality
m=
∞
X
dk qk ,
k=1
by writing
m =: [d1 d2 d3 . . . ].
To give a simple example, qi has only one non-zero element in its encoding,
qi = [0 . . . 010 . . . ]
where the 1 appears as the ith digit, di = 1.
It is important to keep in mind that not every sequence (dk )∞
k=1 of digits satisfying
dk ≤ ak+1 forms a β-expansion of a natural number. In particular, the sequences of
digits satisfy the following requirement [48, 45].
Fact 15 Let [d1 d2 d3 . . . ] be the β-expansion of a positive integer m. For n ∈ N, if
dn 6= 0 then dn+1 6= an+2 .
Proof of 15:
Assume that there is an n ∈ N with dn 6= 0 and dn+1 = an+2 . Since dn+1 6= 0,
134
this digit dn+1 was generated in some step i of the construction of the expansion so
that mi =
Pn+1
k=0
dk qk . By looking at the last two terms of this sum we see that
qn+2 = qn+1 an+2 + qn ≤
n+1
X
dk qk = mi ,
k=1
which contradicts the fact that the qn+1 was “greedily” taken as the largest qk ≤ mi ,
i.e., n + 1 6= max{k : qk ≤ mi }.
2
We are ultimately interested finding a bound on the rotations Rβm where m =
[0 . . . 0dj dj+1 . . . ] and dj 6= 0. Since our rotations are additive in the sense that,
d q
d
Rβm = Rβj j ◦ Rβj+1
qj+1
d
◦ Rβj+2
qj+2
◦ ...
we will be able to break apart the single rotation into a series of ”nice” rotations and
find
Fact 16 Given a rotation, Rβ , on T by β =
cf [a1 a2 a3
. . . ], where the denominators
of the convergents are given by (qi ), if n = [0 . . . 0dj dj+1 . . . ] with j > 1 and dj 6= 0
then
kxn k = kRβn (0)k > C2 /qj+1 ,
where C2 =
1
.
max(an ) + 2
We come back to the proof of Fact 16 after establishing the following bounds on
d qj
“nice” values of n. From Fact 2 we can see that Rβj
moves points to the right if
and only if j is even. Thus, for m = [0 . . . 0dj dj+1 . . . ] with j > 1, it is clear that Rβm
135
will achieve a maximum rotation to the right by setting dk = 0 for all odd k, and
dk = ak+1 for all even k ≥ j. Similarly for an odd j and a left rotation. The following
fact gives a bound on these ”maximal” rotations.
Fact 17 Given j > 1 if for each n ∈ N we define m(n) = [d1 d2 . . . ] where for each
i ∈ [0, 1, 2, . . . , n] we set dj+2i = aj+2i+1 and dk = 0 for all other k ∈ Z, then
m(n)
lim Rβ
n→∞
−qj−1
(x) = Rβ
(x).
Proof of Fact 17:
We start by using induction to show that −qj−1 = m(n) − qj+2n+1 . At n =
0 this is merely a rewriting of qj+1 = aj+1 qj + qj−1 . For n ≥ 1 we assume that
−qj−1 = m(n − 1) − qj+2(n−1)+1 . Then using that qj+2n+1 = aj+2n+1 qj+2n + qj+2n−1
and m(n − 1) =
Pn−1
i=0
aj+2i+1 qj+2i to rewrite
−qj−1 =
=
n−1
X
i=0
n
X
aj+2i+1 qj+2i
!
+ aj+2n+1 qj+2n − qj+2n+1
aj+2i+1 qj+2i
!
− qj+2n+1
i=0
= m(n) − qj+2n+1
So from the additivity of the rotation we have
m(n)
Rβ
−qj+2n+1
◦ Rβ
−qj−1
(x) = Rβ
(x).
Thus to verify (17) all that is left is to show that
−qj+2n+1
Rβ
(x) → x as n → ∞
136
Rβ is an isometry so
−qj+2n+1
x − Rβ
q
q
q
(x) = Rβj+2n+1 (x) − x = Rβj+2n+1 (0) − 0 = Rβj+2n+1 (0) = xqj+2n+1 .
So we will have
−qj+2n+1
Rβ
(x) → x if and only if xqj+2n+1 → 0.
From the construction of our continued fraction we already know that kxqj+2n+1 k <
kxqj+2n k so (kxqj+2n+1 k)∞
n=0 forms a decreasing sequence, however we need to preclude
the possibility of the sequence limiting at a non-zero value. To do so we observe that
for any j
a
q
q
xqj+1 = Rβj+1 j ◦ Rβj−1 (0) = xqj−1 + aj+1 xqj ,
which, since xqj and xqj+1 have opposite signs, we use to write
kxqj−1 k = kxqj+1 k + aj+1 kxqj k
(4.14)
≥ kxqj+1 k + kxqj k
(4.15)
≥ kxqj+1 k + kxqj+1 k = 2kxqj+1 k.
Applying this n + 1-times to 1 ≥ kxqj−1 k we see that 1 ≥ 2n+1kxqj+2n+1 k, so
lim kxqj+2n+1 k = lim
n→∞
−qj+2n+1
and as a consequence Rβ
a
n→∞ 2n+1
=0
(x) → x as n → ∞ and
m(n)
lim Rβ
n→∞
−qj−1
(x) = Rβ
(x)
as desired.
2
137
To prove fact 16 we will also make use of the following well know fact, which
is a special case of a conjecture by Steinhaus (also known as the 3-gap theorem or
3-distance theorem) proofs of which followed by many authors, [49] and [50] represent
2 of the earlier proofs. In the general statement of the Steinhaus conjecture one looks
at the distances between successive points in the set (xn )N
n=0 . If xp , x0 = 0 and xa are
the three successive points (prior to, at , and after zero), then the distances between
any two successive points are kxp k, kxa k or kxp k + kxa k with the last distance only
possible for N < p + a − 1. Our special case will have N = p + a − 1.
q
j+2
Fact 18 {xn }n=0
−1
partitions T into qj+2 segments each of magnitude kxqj+1 k +
kxqj+2 k or kxqj+1 k.
Proof of Fact 18:
In (the example following Lemma 1.3 in) [51] it is shown that (for N = qj+2 +
qj+1 − 1)
qj+1 kxqj+2 k + qj+2kxqj+1 k = 1.
Without loss of generality we will assume that xp = xqj+2 and xa = xqj+1 . We wish
to decrease N by qj+1 , as we do so a third distance kxqj+2 k + kxqj+1 k is introduced.
Each point we remove combines two intervals into one and from Proposition 4.2 in
[52] we know that the new interval will have a length equal to the largest of the three
possible lengths – that is, of length kxqj+2 k + kxqj+1 k. For each point we remove (until
xp or xa changes) we gain a gap of size kxqj+2 k + kxqj+1 k and, necessarily, lose one
gap of size kxqj+2 k and another of size kxqj+1 k. These three distances persist until N
138
decreases by qj+1 , at which time all gaps of size kxqj+2 k have been removed (at that
point also xqj+2 is no longer part of the collection (xn )N
n=0 and we must have a new
xp ) leaving us with qj+2 + qj+1 − qj+1 = qj+2 gaps of size kxqj+1 k + kxqj+2 k or kxqj+1 k.
2
The changing gap structure in the proof is summarized by the following computation where we remove qj+1 intervals of size kxqj+2 k and kxqj+1 k and replace them
with qj+1 intervals of size kxqj+2 k + kxqj+1 k.
1 = qj+1 kxqj+2 k + qj+2 kxqj+1 k
= qj+1 kxqj+2 k + (qj+2 − qj+1 )kxqj+1 k + qj+1 )kxqj+1 k
= qj+1 kxqj+2 k + qj+1 )kxqj+1 k + (qj+1(aj+2 ) + qj − qj+1 )kxqj+1 k
= qj+1 (kxqj+2 k + kxqj+1 k) + (qj+1(aj+2 − 1) + qj )kxqj+1 k
(4.16)
Since (qj+1 (aj+2 − 1) + qj ) = qj+2 − qj+1 (4.16) expresses the desired number
of gaps, each of the length we expected from the proof of Fact 18. Our new xp
must then havekxp k = kxqj+1 k + kxqj+2 k. Indeed, from [52] (which examines gap
structure) or from our continued fraction construction we can see that this new xp =
(a
Rβ j+2
−1)qj+1
(xqj ) = xqj+1 (aj+2 −1)+qj and has magnitude kxqj k − (aj+2 − 1)kxqj+1 k. By
(4.14)
kxqj k − (aj+2 − 1)kxqj+1 k = kxqj+1 k + kxqj+2 k,
so kxp k = kxqj+1 k + kxqj+2 k.
Proof of Fact 16:
139
Without loss of generality assume that xqj is in the right interval, as a consequence all xqj +2i are in the right interval and xqj +2i+1 are in the left. We look at Rβn
d q
for n = [0 . . . 0dj dj+1 . . . ] as three rotations; a rotation by Rβj j , then the rotation to
the right by Rβr with r = [0 . . . 0dj+2 0dj+40 . . . ], and a rotation to the left by Rβl with
l = [0 . . . 0dj+10dj+30 . . . ]. From fact 17 the magnitude of the rotation Rβr is less than
q
the magnitude of the rotation given by Rβj+1 so
q
kRβr (0)k < kRβj+1 (0)k = kxqj+1 k.
q
Similarly the rotation Rβl is less than the magnitude of the rotation given by Rβj ;
however since dj > 0 by 15 we have dj+1 6= aj+2 and we can find a better approximation. Since dj+1 6= aj+2 our rotation to the left is assured to be at least the magnitude
q
of one rotation by Rβj+1 less than the maximal left rotation (from Fact 17) and
q
q
kRβl (0)k < kRβj (0)k − kRβj+1 (0)k = kxqj k − kxqj+1 k.
d q
We can now examine the location of xn = Rβn (0) = Rβl (Rβr (Rβj j (0))) for dj ∈
[1, 2, . . . aj+1 ]. From the bounds on the right and left rotations we know that
xn ∈ (xdj qj − (kxqj k − kxqj+1 k), xdj qj + kxqj+1 k).
Notice (see Figure 15) that this interval has a diameter of kxqj k and that the intervals
for di and di + 1 (when di + 1 ≤ aj+1 ) share an endpoint. So the union over all
1 ≤ di ≤ aj+1 is contained in (kxqj+1 k, kxqj+1 k + kxqj k). From (4.15) we recognize that
(kxqj+1 k, kxqj+1 k + kxqj k) ⊂ (kxqj+1 k, kxqj−1 k). Since kxqj+1 k < kxqj−1 k ≤ kβk < .5 we
know that kxn k > min{kxqj+1 k, .5} = kxqj+1 k.
140
The 3 regions of size kxqj k that could contain xn
q
q
Rβj
Rβj
xqj−1
q
Rβj
xqj+1 0
xqj
x2qj
x3qj
kxqj+1 k
kxqj−1 k
Figure 15. Shown here (with aj+1 = 3) are three regions, one of which contains xn .
The leftmost edge of the union of the three boundary regions is a distance kxqj+1 k
from zero. The right edge is a distance equal to min{kxqj−1 k, 1 − kxqj−1 k}, both of
which are longer than kxqj+1 k.
q
−1
j+2
To relate kxqj+1 k directly to qj+1 we partition T by {xn }n=0
. Fact 18 assures
us that this partition has segments of lengths kxqj+1 k and kxqj+1 k + kxqj+2 k. In fact
we know (4.16)
1 = qj+1(kxqj+2 k + kxqj+1 k) + (qj+1 (aj+2 − 1) + qj )kxqj+1 k
Replacing all segments of length kxqj+2 k with (the longer) segments of length kxqj+1 k
141
we see that
1 < qj+1(kxqj+1 k + kxqj+1 k) + (qj+1 (aj+2 − 1) + qj )kxqj+1 k
= qj+1(kxqj+1 k) + (qj+1 (aj+2) + qj )kxqj+1 k
= (qj+1 + qj+2)kxqj+1 k
≤ (qj+1 + qj + aj+2 qj+1 )kxqj+1 k
< (qj+1 + qj+1 + aj+2 qj+1)kxqj+1 k
= (2 + aj+2 )qj+1 kxqj+1 k.
So,
kxqj+1 k >
Thus, kxn k > kxqj+1 k >
1
(2 + aj+2 )qj+1
≥
1
.
(2 + maxn (an ))qj+1
1
.
(2 + maxn (an ))qj+1
2
142
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