Rotation Sets and Entropy
A Dissertation Presented
by
Jaroslaw Marcin Kwapisz
to
The Graduate School
in Partial Fulllment of the Requirements
for the Degree of
Doctor of Philosophy
in
Mathematics
State University of New York
at
Stony Brook
August 1995
State University of New York
at Stony Brook
The Graduate School
Jaroslaw Marcin Kwapisz
We, the dissertation committee for the above candidate for the Doctor of
Philosophy degree, hereby recommend acceptance of the dissertation.
John Milnor
Distinguished Professor of Mathematics
Dissertation Director
Yair Minsky
Assistant Professor of Mathematics
Chairman of Defense
Mikhail Lyubich
Associate Professor of Mathematics
Folkert Tangerman
Visiting Assistant Professor
Department of Applied Mathematics and Statistics
Outside Member
This dissertation is accepted by the Graduate School.
Graduate School
ii
Abstract of the Dissertation
Rotation Sets and Entropy
by
Jaroslaw Marcin Kwapisz
Doctor of Philosophy
in
Mathematics
State University of New York at Stony Brook
1995
This thesis is concerned with the dynamics generated on the
two dimensional torus by a homeomorphism isotopic to the identity. For each rotation vector of such a map, we dene the topological entropy at the rotation vector ; this is a function having the
rotation set of the map for its natural domain and the topological entropy of the map for its maximum. We show how the ideas
of the thermodynamical formalism aid the evaluation of this function. For maps that are pseudo-Anosov (relative to a nite set),
we demonstrate its strict convexity and real analyticity. We then
describe those Gibbs states that realize the topological entropy at
the rotation vector as their measure theoretic entropy.
iii
In the general case, we show that the topological entropy is
positive at the rotation vector that lies in the interior of the rotation
set. We prove that any such vector is, in fact, the common rotation
vector for points of a compact invariant set that carries positive
topological entropy.
Presumably, the entropy on this set can be made arbitrarily
close to the topological entropy at the corresponding rotation vector. We verify this for pseudo-Anosov maps and prove that, in general, both entropies are explicitly estimated from below in terms of
the size of the rotation set and the relative distance of the vector
from the boundary. A key tool here is the theory of quasi-conformal
mappings.
Finally, we analyze the rotation sets arising in a model of a
resonantly kicked charged particle in a constant magnetic eld.
This provides a physically interesting example to which our general
theorems apply.
iv
To Agnieszka; To the wind
Contents
List of Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii
Acknowledgements : : : : : : : : : : : : : : : : : : : : : : : : :
ix
Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
1
1 An estimate of entropy for toroidal chaos : : : : : : : : : : : 11
1.1 Statement of the result : : : : : : : : : : : : : : : : : : : : : :
11
1.2 Comparison with pseudo-Anosov maps : : : : : : : : : : : : :
14
1.3 Entropy and conformal structures : : : : : : : : : : : : : : : :
16
1.4 Proof of the Main Lower Bound : : : : : : : : : : : : : : : : :
19
2 Topological entropy at the asymptotic average : : : : : : : : 28
2.1 Relative variational principle and thermodynamical formalism
29
2.2 Topological entropy at the asymptotic average for subshifts of
nite type : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
44
2.2.1 Transfer matrix, Gibbs states, and explicit formulas : :
45
2.2.2 Observable shadowing with the right entropy : : : : : :
53
2.2.3 The scaold of subshifts of nite type : : : : : : : : : :
61
vi
3 Topological entropy at the rotation vector for torus maps
68
3.1 Denitions and the main result : : : : : : : : : : : : : : : : :
3.2 The central example: pseudo-Anosov maps : : : : : : : : : : :
69
72
3.3 Proof of the main result : : : : : : : : : : : : : : : : : : : : :
77
3.4 Example : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
81
4 Rotation sets for the resonantly kicked linear oscillator : : 87
4.1
4.2
4.3
4.4
The physical context : : : : : : : : : : : : : : : : :
The mapping and its rotation set : : : : : : : : : :
The recurrence, higher period orbits, monotonicity :
A mechanical toy-model : : : : : : : : : : : : : : :
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
: 87
: 90
: 94
: 100
A Elementary large deviations for topological dynamics : : : 102
Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 112
vii
List of Figures
3.1
3.2
3.3
3.4
When a and b are not antipodal.
The map h# on the spine. : : : :
The ecient map g1 . : : : : : : :
The transition graph. : : : : : : :
viii
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
71
82
83
84
Acknowledgements
I am indebted a great deal to my adviser John Milnor and to Philip
Boyland for their support during my four years at Stony Brook.
I would also like to thank Michal Misiurewicz and Feliks Przytycki whose
teaching and mentoring began my adventure with dynamical systems.
I acknowledge the Alfred Sloan Foundation for supporting this work through
the doctoral dissertation fellowship.
I owe special thanks to Agnieszka Bielinska, Benjamin Hinkle, and Albert
Fisher for many remarks that improved this text, and to Scott Sutherland and
Brian Yarrington for their help with the computer typesetting.
ix
Introduction
This work originated from an attempt to understand certain aspects of
dynamics generated on a two-dimensional torus by a homeomorphism that is
isotopic to the identity. Our main object of study is the interplay between the
rotation set and the topological entropy. We put these two classical invariants
in the same context by using the ideas of the thermodynamical formalism, and
we nd a qualitative relation between them by invoking the methods of the
theory of quasi-conformal mappings.
In what follows, we introduce the basic concepts, advertise our main results, and show how they are a continuation of the work by others. We tend
to be brief since a signicant amount of expository material can be found in
the subsequent chapters.
Our favorite model of a two-dimensional torus T2 is that obtained by
identifying points of R2 which dier by a vector having integer coordinates.
The resulting projection : R2 ! T2 is a universal covering and we x it once
and for all. The space of all homeomorphisms of T2 that are isotopic to the
identity will be denoted by H(T2 ). If f : T2 ! T2 is in H(T2) then it lifts to
F : R2 ! R2 which commutes with the deck group, i.e. F (~x + v) = F (~x) + v
1
2
for all x~ 2 R2; v 2 Z2 ; we will write H~ (T2) for the space of all lifts of elements
of H(T2 ).
Note that a map F 2 H~ (T2) is a C 0-bounded perturbation of the identity
by ~F = F id : R2 ! R2 which, being invariant under precompositions
with the deck translations, descends to a continuous function F : T2 !
R2 that is called the displacement of F . Thus, for x~ 2 R2 and x = (~x),
F n(~x) x~ = F (x) + ::: + F (f n 1(x)) and the set of all limit points of the
sequence ( n1 (F n(~x) x~))n2N is bounded. This set is the rotation set of x~ under
F , denoted also by (F; x~).
For a point x 2 T2, the set (F; x~) does not depend on the choice of
x~ 2 1(x), so we can also write (F; x). In the case when (F; x) consists
of one point we call it the rotation vector of x. This happens, for example,
when x is periodic. More generally, if is an ergodic probability measure for
f and x 2 T2 is typical for , then the Birkho ergodic theorem implies that
R
(F; x) = f F dg.
The denition of (F; x) is in the spirit of Poincare's notion of the rotation number for a circle homeomorphism | for denitions and historical
background one may consult [ALM93, Boy92, MZ89]. Much as in the case of
degree-one circle maps ([NPT83]), the set (F; x~) generally changes as we vary
x~. Unfortunately, the union Sx~2R (F; x~) often fails to be compact. For this
reason, following [MZ89], we dene the rotation set of F as
!
[ 1 n
\
2
(F ) := cl
(F (~x) x~) : x~ 2 R R2:
n
n>m
m>0
2
Without going into details, let us just note that other existing denitions of
3
the rotation set yield sets that are contained in the above and may at most lack
some points in the boundary. (These can not be the extremal points either.)
As the intermediate value theorem forces the rotation set for a degree-one
circle map to be an interval ([NPT83]), the following result is ultimately a
consequence of the Jordan simple curve theorem.
Theorem 0.0.1 ([MZ89]) If F 2 H~ (T2) then (F ) is convex.
Let us note that, while the denition of (F ) makes sense for non-invertible
maps F , it renders then a set that is a continuum but often lacks convexity
([MZ89]).
Theorem 0.0.1 makes one ask: Can any compact convex subset of R2 be
realized as (F ) for some F 2 H~ (T2 )? This question is still open. Even
deciding which line segments are rotation sets is not yet resolved. Given any
interval with rational slope passing through a point with rational coordinates
it is easy to nd a homeomorphism that has the interval as its rotation set.
The construction in [FM90] implies that intervals with at least one rational
endpoint and irrational slope can be realized as well. Based on their results
for time-one maps of toral ows, Franks and Misiurewicz conjecture that these
are the two only possibilities ([FM90]). In the case of nonempty interior, it
is known that all polygons with vertices having rational coordinates can be
realized as rotation sets ([Kwa92]). There is also an example with (countable)
innity of the extremal points ([Kwa]).
While the basic nature of the rotation set is not fully understood, there is
a considerable number of results linking it to dynamical phenomena exhibited
4
by the map. The following theorem of Franks is a pillar of the progress in this
direction.
Theorem 0.0.2 ([Fra89]) If F 2 H~ (T2) and p 2 Z2 , q 2 N are such that
p=q 2 int((F )), then F q (~x) = x~ + p for some x~ 2 R2. Moreover, if q and the
two coordinates of p have no common divisor, then q is the least j 2 N with
the property that F j (~x) x~ 2 Z2 .
Remark. If v is an extremal point of (F ), then v = R F d for some ergodic
measure ([MZ89]). Thus the assertion of the theorem is true also when p=q
is extremal, by Theorem 3.5 in [Fra88].
In terms of the map f 2 H(T2 ) that is a projection of F , the above
theorem says that any vector v in the interior of (F ) with rational coordinates
is the rotation vector of a periodic point x = (~x) 2 T2 . Moreover, the period
of this periodic point can be required to be equal to the smallest denominator
q in the representation v = p=q, where p 2 Z2 , q 2 N. (Such periodic points
are often called primitive.) Once again the assumption about invertibility of f
is relevant; Barge and Walker have a non-invertible example with no periodic
points and with the nonempty interior of the rotation set ([BW93]).
One reason why Franks's result is important is that it facilitates application of very powerful methods of the Thurston-Nilsen theory ([Thu88, FLP79,
Boy94]) to f 2 H(T2) with nonempty interior of the rotation set. Indeed,
by puncturing T2 at the points of the orbits found by Theorem 0.0.2, one
gets a surface of negative Euler characteristic with a homeomorphism that,
not only does not have to be isotopic to the identity, but often is isotopic to
5
a pseudo-Anosov map (cf. Theorem 1.2.1). This trick, going back to Bowen
([Bow78a]), was applied by Llibre and MacKay and subsequently by Misiurewicz and Ziemian to prove, among other things, the following theorems.
Theorem 0.0.3 ([LM91]) If f 2 H(T2), F 2 H~ (T2) is its lift, then, for any
continuum C int((F )), there is a point x 2 T2 with (F; x) = C .
Theorem 0.0.4 ([MZ91]) If f 2 H(T2) and F 2 H~ (T2) is its lift, then, for
any v 2 int((F )), there is an invariant set K T2 such that (F; x) = v for
all x 2 K .
Thus nonempty interior of the rotation set implies uncountably many
invariant sets each with a dierent non-collinear rotation vector. Analogous
behavior for annulus homeomorphisms is exhibited by integrable twist maps;
however, on the torus (as with degree-one circle maps), it is a sure sign of
chaos.
Theorem 0.0.5 ([LM91]) If f 2 H(T2), F 2 H~ (T2 ) is its lift and int(F ) 6=
;, then the topological entropy htop (f ) of f is positive.
For degree-one circle maps there exist explicit and sharp lower bounds for
the topological entropy in terms of the rotation set ([ALM93]); this prompted
Llibre and MacKay to ask whether there is such a relation in the case of T2.
We give an answer to this question | a precise formulation of the result can
be found in the beginning of Chapter 1. Below, we give a simplied version
that is formally still a conjecture but approximates our theorem very well.
6
\Theorem" 1 (cf. Theorem 1.1.1) There exist a universal constant C > 0
such that, if f 2 H(T2 ) and F 2 H~ (T2) is its lift, then
(i) htop (f ) C ln Area((F )), for (F ) that are not \small";
q
(ii) htop (f ) C Area((F )), for (F ) that are not \large".
Our estimate is not sharp. (Nevertheless, it captures the right asymptotics
for \large" rotation sets | see the example in Chapter 1.) We think of it more
as an evidence of the power of our method than an isolated result. The method
that we advocate takes over where Llibre and Mackay left o and exploits the
connection between the topological entropy and the quasi-conformal dilatation
of the underlying pseudo-Anosov map. This approach has a lot of potential
and ultimately should lead to better results.
The second main idea in this thesis is that of topological entropy at the
rotation vector. Once again our motivation was Theorem 0.0.5 which made us
ask: If int((F )) 6= ;, how much topological entropy is contributed by the points
with a particular rotation vector? We make the rst step towards answering
this question. Namely, for a map f 2 H(T2 ) and its lift F 2 H~ (T2), we dene
for each v 2 (F ) the topological entropy at the rotation vector, denoted by
(v)
(v)
htop
(F ). We show that htop
(F ) obeys a form of the thermodynamical formalism, which we then employ to give a complete analysis of the pseudo-Anosov
case and, ultimately, strengthen \Theorem" 1 to the following:
\Theorem" 2 (cf. Theorem 3.1.1) For some universal constant C > 0, if
f 2 H(T2 ) is a dieomorphism, F 2 H~ (T2) is its lift and v 2 int((F )), then
there exists a compact invariant set K T2 such that (F; x) = v, for all
7
x 2 K , and
(v)
htop
(f ) htop (f jK ) C ln Area((F )) (v);
for (F ) that are not \small",
q
(v)
htop
(f ) htop (f jK ) C Area((F )) (v);
for (F ) that are not \large".
The quantity (v) is a certain measure of the \distance" of v from the boundary
of (F ).
Ultimately, one should be able to make the choice of the set K above so
(v)
(f ). We prove this only for maps
that htop (f jK ) is arbitrarily close to htop
that are pseudo-Anosov relative to a nite set (see [Boy94] for the denition).
Another important issue, which we resolve here only for that class of maps,
(v)
is the regularity of htop
as a function of v. We have the following theorem
(summarizing Proposition 3.2.2 and Theorem 3.2.1).
Theorem 0.0.6 If f 2 H(T2) is pseudo-Anosov relative to a nite set and
F 2 H~ (T2) is its lift, then
(v)
(i) htop
(F ) is real-analytic and strictly concave on int((F ));
(ii) for any v 2 int((F )), there exists a fully supported ergodic measure (v)
R
(v)
such that the measure theoretic entropy h v (f ) = htop
(F ) and F d(v) = v;
( )
(iii) for any v 2 int((F )) and 2 (0; 1), there exists a compact invariant
(v)
subset K of T2 such that htop (F jK ) htop
(F ) and (F; x) = fvg for all
x 2 K.
8
Actually, there is more in our analysis than the above theorem spells out; for
(v)
example, htop
(F ) can be explicitly calculated from the Markov partition.
The line of attack in proving the theorem is not original at all: we use
symbolic dynamics to reduce it to facts about subshifts of nite type. Still, our
discussion of subshifts of nite type contains some new results. In particular,
the following theorem may be useful in many applications.
Theorem 0.0.7 (cf. Theorem 2.2.4) Suppose : ! is a transitive subR
shift of nite type and : ! Rd is locally constant. If v = d for some
invariant ergodic probability measure , then, for any 2 (0; 1), there exists a
compact invariant set K such that htop ( jK ) h ( ) and
nX1
i
(f (x)) nv < +1:
sup n2N;x2K i=0
The above theorem for integer valued and integer v coincides with the re-
sult in [MT92]. There, it is applied to calculate the minimal entropy of some
sequences of patterns of periodic orbits for one-dimensional maps. The discussion in [MT92] yields concrete formulas only for symmetric patterns. Our
methods (which are dierent) can be used to treat equally eciently the general case. In particular, the Gibbs states that we discuss realize the variational
principle in [MT92] thus answering the question posed there. 1
(v)
As a tool for comprehending htop
we develop (in Chapter 2) a weak but
quite general version of the thermodynamical formalism for a dynamical system on a compact space with a continuous observable in Rd. Theorem 2.1.1
1 The
relation of my work to that of [MT92] was brought to my attention only
recently. I will give the details elsewhere.
9
(v)
plays the central role here by connecting the entropy htop
with the appropriate
pressure function. Even though this theorem can be viewed as a consequence
of the standard thermodynamical formalism ([Wal82]), the implication is not
trivial; to establish it, we generalize the variational principle for topological
entropy ([Wal82]) by making it sensitive to the averages of the observable (see
Theorem 2.1.2). This material should be thought of as just another realization
of the paradigm of the thermodynamical formalism ([Rue78, Bow75, Wal82]).
We hope that our presentation will help to absorb these ideas into the theory of rotation sets (including the context of other surfaces than torus, see
[Fra94, Boy94].)
(v)
Let us also note that in dealing with htop
one can get pretty far using
elementary large deviation techniques. The inspiration for this approach comes
from a book by Ellis ([Ell85]). In Appendix A we use it to give a proof of the
thermodynamical formalism theorem (Theorem 2.1.1).
Finally, our last chapter is devoted to a concrete family of torus homeomorphisms that are isotopic to the identity. The purpose of this discussion is two-fold. For one, we wanted to give a concrete physical model that
satises the assumptions of our theorems. The other reason is mathematical: these are perhaps the simplest and most accessible analytically mappings exhibiting nonempty interior of the rotation set; even though studied from other points of view for quite some time, they somehow avoided
to this date theoretical analysis with the emphasis on their rotation sets
([LL93, ZZRSC86b, CSUZ87, ZZRSC86a]). We give the beginning of such
10
an analysis. In particular, we prove monotonicity of the rotation set for a
certain (physically) natural one-parameter family of torus maps. This may be
interesting since such examples were not known before.
Chapter 1
An estimate of entropy for toroidal chaos
In this chapter we show that topological entropy can be estimated from
below by looking at the \size" of the rotation set. This \size" can be roughly
thought of as the area of the rotation set (but, in general, is not comparable
with it).
1.1 Statement of the result
Consider f 2 H(T2 ) and its lift F 2 H~ (T2). Assume that int((F )) 6= ;.
It was proved in [LM91] that f exhibits then toroidal chaos; in particular, f
has positive topological entropy htop (f ) > 0 (see Theorem 0.0.5). Our goal is
to provide a lower bound on the entropy in terms of the size of the rotation
set. To formulate the result, let us rst explain how we measure the size of
the rotation set.
11
12
Denition 1.1.1 Let K R2 be convex and compact. If int(K ) contains
three non-collinear points of Z2 , then we dene
A(K ) := maxfpr r : r ; r 2 N and there exist p; v; w 2 Z2 such that
1 2
1 2
p; p + r1v; p + r2w 2 int(K ) and are non-collinearg;
I (K ) := 1:
Otherwise, we set
A(K ) := 0;
I (K ) := inf f 2 N : int(K ) contains three non-collinear points of Z2 g:
Note that, for m 2 N, if A(K ) > 0, then A(mK ) mA(K ), and that
A ( I ( K ) K ) 1:
(1:1)
Theorem 1.1.1 (Main Lower Bound ([Kwa93])) There exist universal constants C , C 0 such that, for any f 2 H(T2 ) and its lift F 2 H~ (T2 ), we have
htop (f ) C ln+ A((F ));
htop (f ) C 0=I ((F )):
We think of the rst estimate as signicant for \large" (F ) and of the second
as signicant for \small" (F ).
Remark. In fact, we will prove that if r1 and r2 realizing A((F )) are both
greater than 5 then
q
htop (f ) 31 ln (r1 1)(r2 1) 12 ln 4:
13
By considering the eighth iterate of F , this yields C = 1=27 and C 0 = ln 2=54
| see Claim 1.4.1 and the discussion opening Section 1.4.
As indicated by the remark, our statement of the theorem is a compromise
between sharpness of the result and complexity of its formulation. It will be
clear from the proof that we are far from providing optimal estimates, so we
see no point in complicating the inequalities. Nevertheless, the theorem is
asymptotically sharp (up to a multiplicative constant) for \large" rotation
sets, as shown by the following example.
Example. For n 2 N, dene Un ; Vn 2 H~ (T2) by Vn(x; y) := (x+n sin(2y); y),
Un(x; y) := (x; y + n sin(2x)), x; y 2 R. Consider Fn := Vn Un 2 H~ (T2) and
the corresponding fn 2 H(T2). The points (0; 0); ( 41 ; 0); (0; 14 ) have rotation
vectors (0; 0); (0; n); (n; 0) respectively, so A((Fn )) n. On the other hand,
the maximal Lyapunov exponent dominates htop (fn) and is trivially bounded
by ln sup krfnk ln(2n) . In this way, we have
lim sup ln (htop (fn)=A((Fn))) 1:
n!1
One would like to have an example of a family ffg>0, f 2 H(T2),
with lifts fFg>0 having (F) shrinking to a point, as ! 0, and such that
htop (f) I ((F)) Const, for > 0. This would mean that our estimate is
asymptotically sharp for \small" rotation set. No such example is known to
the author.
The rest of this chapter is devoted to the proof of the theorem. We isolate
the main steps into separate sections. Section 1.2 describes reduction to the
14
case of a pseudo-Anosov map. Section 1.3 explains how entropy is linked
with the quasi-conformal dilatation of conformal structures on T2 . The very
estimates proving the theorem can be found in Section 1.4.
1.2 Comparison with pseudo-Anosov maps
We describe here a trick that allows one to pass from investigation of a
general map in H(T2) with nonempty interior of the rotation set to a pseudoAnosov map with approximately the same rotation set. Our version is only a
trivial extension of the original one in [LM91]. We use the notion of a pseudoAnosov map (of a surface) rel a nite invariant set. Referring the reader to
[Boy94] for an exposition, we only say here that, if f : S ! S is a pseudoAnosov map rel a nite invariant set P , then it is a factor of a pseudo-Anosov
map ([Thu88, FLP79]) on the surface obtained by compactifying the ideal
boundary of S n P with circles. The factor map collapses the boundary circles
to the corresponding points of P and is injective otherwise. Thus the dynamics
of the two maps almost coincide. (In particular, they have the same topological
entropy ([Bow78a]).)
Theorem 1.2.1 Suppose that F 2 H~ (T2) and int((F )) 6= ;. Let f 2 H(T2)
be the map that lifts to F . If 0 int((F )) is a convex non-degenerate polygon
with vertices in Q2 , then there exists a nite f -invariant set P T2 and a
pseudo-Anosov rel P map g 2 H(T2 ) that is isotopic to f rel P and has a lift
G 2 H~ (T2) with 0 (G).
15
Remark. The above theorem is also true when 0 has some vertices on the
boundary of (F ) as long as these are extremal points of (F ) (cf. the remark
after Theorem 0.0.2).
Pseudo-Anosov maps have minimal topological entropy in their isotopy
class ([Thu88, FLP79]), so
htop (f ) htop (g) > 0:
(1:2)
Remark. Pseudo-Anosov maps not only minimize entropy but also have \the
simplest dynamics" in their isotopy class ([Han85, Boy10]). In particular, we
have (G) (F ). (For details and proofs see Theorem 3.3.1 and the corollary
after it.)
Proof of Theorem 1.2.1. Let v1 ; :::; vm 2 Q2 be the vertices of 0 (or, equally
well, any nite set in Q2 such that 0 = conv fv1; :::; vm g, where conv abbreviates convex hull). Write vi = pi=qi, where pi 2 Z2 , qi 2 N and the two
components of pi and qi have no common divisor, all this for i = 1; :::; m.
By Theorem 0.0.2, there exist x1 ; :::; xm 2 R2 such that F qi (xi ) = xi + pi,
i = 1; :::; m. Let P be the union of the projections on the torus of the orbits
of x1 ; :::; xm . If g 2 H(T2) is isotopic to f rel P and G 2 H~ (T2) is the lift of g
equivariantly isotopic to F relative 1(P ), then 0 = convfv1 ; :::; vmg (G).
In particular, we can take for g the Thurston canonical representative of the
isotopy class of f relative P (see e.g. Th. 7.1 in [Boy94]). Following [LM91],
one can argue that g can not have any reducing curves, and so it is pseudoAnosov. Indeed, since a reducing curve must be non-peripheral, we have a
priori three possibilities: it is essential on T2 , and then (G) would have to
16
be contained in a line; it bounds a disk containing points of dierent orbits
in P , which would contradict the fact that the rotation vectors of these orbits
are dierent; it bounds a disk containing two or more points of one orbit in
P , which would force pi and qi to have a common divisor for some i, contrary
to the assumption.
Q.E.D.
1.3 Entropy and conformal structures
It is clear from the previous section that, to prove the Main Lower Bound,
we merely have to correlate the size of the rotation set with the topological
entropy for pseudo-Anosov maps. A natural rst impulse is to try linking the
two via standard symbolic dynamics given by the Markov partitions. This
however leaves one with a task of understanding how topology of the torus
inuences the structure of the corresponding subshifts of nite type. We propose here a dierent approach. Namely, we will take advantage of the fact that
the the entropy of the pseudo-Anosov map g can be calculated by looking at
its quasi-conformal dilatation with respect to a preferred family of conformal
structures on T2. We explain this below. (Following [Alh66], we abbreviate
quasi-conformal to q.c.)
Denote by C (T2 n P ) the set of all conformal structures on T2 n P . If
2 C (T2 n P ) and : T2 n P ! T2 n P is a -q.c. map, then we write K [ ]
for the (maximal) q.c. dilatation of ([Alh66]). (For conformal maps K = 1.)
According to Bers (Ch. III, [Abi80]), there exists g 2 C (T2 n P ) solving the
17
following extremal problem
Kg [g] = inf fK [ ] : 2 C (T2 n P );
isotopic to gg:
(1:3)
Moreover, g is modeled around each boundary component of T2 n P on a
punctured disc (see Th. 3, x2, Ch. III, [Abi80]). Thus g extends uniquely
through the punctures to a conformal structure on T2 which, with no risk of
confusion, is also denoted by g .
For the sake of giving a more intuitive picture, let us sketch an alternative (and more geometrical) way of nding g . From the denition of pseudoAnosov map relative a nite set (see [Boy94]), there exists a pair of transverse measured singular foliations F s and F u such that g(F s) = 1F s and
g(F u) = F u, where = exp htop (g). The two foliations dene a smooth
structure together with a at Riemannian metric on the complement of the
set of singularities Q T2 (P Q). This Riemannian metric naturally determines a conformal structure on T2 n Q. However, due to the form of F s and
F u at the singularities that is postulated by the denition, this structure has
removable singularities at points of Q and so determines a unique conformal
structure g on T2. From the action of g on F s and F u one sees the q.c.
dilatation to be ([Abi80])
K [g] = 2 = exp(2htop (g)):
Putting this together with inequality (1.2) we get
h(f ) (1=2) ln Kg [g]:
(1:4)
18
In our applications we will not use any a priori knowledge of g , thus,
weakening (1.4), we state the following theorem.
Theorem 1.3.1 If f 2 H(T2) and P is a collection of primitive periodic
orbits with dierent non-collinear rotation vectors, then
htop (f ) inf f(1=2) ln K [ ] : 2 C (T2 );
-q.c. map of T2 isotopic to f rel P g:
Let us note here a curious problem related to the above theorem.
Open Problem. If S is a compact Riemann surface and f : S ! S is a q.c.
homeomorphism with q.c. dilatation K [f ], is htop (f ) (1=2) ln K [f ] ?
For a C 1-smooth map, the answer is yes. Indeed, by the variational principle for topological entropy ([Man87]), we just need to prove the following:
if is an ergodic invariant measure, then h (f ) ln K [f ]=2. Consider the
Lyapunov exponents + of . By Ruelle's inequality ([Man87]), we can
write h (f ) maxf0; +g and h (f ) = h(f 1) maxf0; g. Thus either h (f ) = 0, and there is nothing to prove, or h (f ) minf+; g (+ )=2. By looking at the derivative Df n(x) at a typical point x and
for large n, we get exp += exp K [f ], which already implies our claim.
In the non-smooth case, there are estimates by Gromov with a non-optimal
multiplicative constant ([Gro87]).
19
1.4 Proof of the Main Lower Bound
We have two inequalities to prove
htop (f ) C ln+ A((F ));
(1:5)
htop (f ) C 0=I ((F )):
(1:6)
and
In fact, (1.6) follows from (1.5). For a proof, set := I ((F )) 2 N. Using
(1.1), we get A((F 2 )) = A(2(F )) 2. Application of (1.5) to f 2 yields
2 htop (f ) = htop (f 2 ) C ln 2. The inequality (1.6) with C 0 = (C ln 2)=2
follows.
In the rest of this section we tackle (1.5). Let p; v; w 2 Z2 and r1 ; r2 2 N
be as in the denition of A((F )). Passing perhaps to the sixth iterate of F , we
may assume that r1 ; r2 6. Also, it is easy to see that we can require that v; w
form a basis of Z2 as a Z-module. By the theorem of Franks (Theorem 0.0.2),
for each of p; p + r1 v; p + r2w 2 (F ) \ Z2 , there is a xed point of f realizing
it as the rotation vector. Let P be the union of those three xed points. By
Theorem 1.3.1, to estimate htop (f ), we need to consider an arbitrary conformal
structure 2 C (T2 ) and a -q.c. map g : T2 ! T2 which is isotopic to f
rel P . By the uniformization theorem ([Abi80]), the conformal torus (T2; )
is biholomorphic to C= , where is a rank-two lattice in C. The map g lifts
to a map : C ! C which is q.c. with respect to the standard conformal
structure on C with the q.c. dilatation K [] = Kg [g]. Let e1 ; e2 2 be the
vectors corresponding to the vectors v; w. After perhaps post-composing 20
with a deck transformation, the three xed points in P lift to p0 ; p1; p2 2 C
such that
(p0 ) = p0; (p1 ) = p1 + r1 e1; (p2) = p2 + r2 e2:
(1:7)
Our task is to verify, from the above topological data on , the following claim
which already implies (1.5).
Claim 1.4.1 (target estimate) If r1; r2 6, then
K [] 41 (r1 1)1=3(r2 1)1=3 :
(1:8)
The proof of the claim is rather technical. To better separate the ideas,
after necessary denitions, we formulate the key observations as separate Facts
and then prove them. All the pieces of the puzzle are put together in the nal
argument at the end of this section.
For any ordered pair (a; b) of orthogonal vectors in C, we have the rectangle R = R(a; b) = fxa + yb : 0 x; y 1g. For any s 2 [0; 1], let
s[R] : [0; 1] ! R be given by s[R](t) = ta + sb, and s[R] : [0; 1] ! R
be given by s[R](t) = sa + tb. Thus we have two families of curves lling R: a-family A := fy [R]gy2[0;1], and b-family B := fx[R]gx2[0;1]. Clearly,
@R = 0[R] [ 1[R] [ 0[R] [ 1[R]. We call the rectangle R fundamental for
if every orbit under of intersects its interior exactly once. All the above
denitions extend in an obvious way to any translate of the rectangle R.
For a family G of rectiable curves in C, we denote the extremal length
of G by (G ), i.e.
ZZ
(G ) := sup L()2 : : C ! R+ measurable with
2dxdy = 1 ;
21
where
L() := inf
Z
jdzj : 2 G :
For the Euclidean length of curve we simply write l( ).
For the families A and B we have ([Alh66])
(A) = jaj2=jRj; (B) = jbj2 =jRj;
(1:9)
where jRj stands for the Euclidean area of R. Our estimation of K [] will be
based on the quasi-invariance of the extremal length, a property encapsulated
by the following inequality ([Alh66])
K [] (G )=(G ):
(1:10)
For G above we will take a-families and b-families of appropriate rectangles. The following simple lemma plays a key role in nding \good rectangles".
Lemma 1.4.1 If v; w 2 C are orthogonal, v 2 and R = R(v; w) is fundamental for , then there exists z 2 C, for which R0 := R + z satises
(A) l(0)2 =jR0j = l(1)2=jR0j;
(1:11)
(B) l(0 )2=jR0j = l(1 )2=jR0j;
(1:12)
where A = fy g, B = fxg are the a-family and b-family for R0 .
Proof. First slide R along the line Rw to nd t0 such that
l((0 [R] + t0 w)) = inf fl((0 [R] + tw)) : t 2 Rg:
22
This means that
l(0[R + z]) = inf fl() : 2 a-family of R + zg;
(1:13)
where z = t0 w. Observe that (1.13) holds also for any z = t0w + sv, s 2 R,
because for any s1 ; s2 2 R the curves y [R + t0 + siv], i = 1; 2, project to the
same loop on the torus and so their images under have equal length. In this
way, if sliding along Rv we have found s0 such that
l(0 [R + z]) = inf fl( ) : 2 b-family of R + zg; z = t0 w + s0 v; (1:14)
we got both (1.13) and (1.14) satised for z = t0w + s0v. Set R0 := R + z.
The inequalities in (1.11) and (1.12) follow from the denition of the extremal
length when one takes for a multiple of the characteristic function of R0
while remembering that jR0j = jR0j. The equalities are trivial since 0 and
1, as well as 0 and 1 , project to the same arcs on the torus.
Q.E.D.
Let 2 [0; ) be the measure of the convex angle between e1 and e2 .
Fact 1.4.1 For i; j 2 f1; 2g, i 6= j , we have
8(jeij2 + sin2 jej j2)K [] (ri 1)2jeij2 + (rj 1)2 sin2 jej j2
+ maxf0; rj jej jj cos j jeijg2:
Proof. Let v := ei and w be the component of ej orthogonal to ei. Let R0 be
as in Lemma 1.4.1. Due to (1.7), (R0) intersects R0 , R0 + riei, R0 + rj ej , so
l((@R0 )) dist(R0; R0 + riei ) + dist(R0 ; R0 + rj ej );
23
and consequently, (using the Pythagorean theorem),
l((@R0 )) (ri 1)jeij
(1.15)
1=2
+ (rj 1)2jej j2 sin2 + maxf0; rj jej jj cos j jeijg2 :
Feeding (1.9), (1.11), (1.12) into (1.10), we obtain
(jeij2 + sin2 jej j2)K [] jeij2 (A)=(A) + jej j2 sin2 (B)=(B)
2
2
jeij2 l(jRj0 ) l(jRj)2 + jej j2 sin2 l(jRj0 ) l(jRj)2
0
0
2
2
= l(0 ) + l(0 )
21 (l(0 ) + l(0))2
= 81 l((@R0 ))2 :
This inequality put together with (1.15) proves Fact 1.4.1.
Q.E.D.
Note that, if the lengths of e1 and e2 were \comparable", Fact 1.4.1 would
yield an estimate similar to (1.8) of Claim 1.4.1. The following fact exhibits
one mechanism preventing the je1 j and je2 j from being too disproportional.
Fact 1.4.2 If d 2 (0; 1) and e1 = 1, e2 = dp 1, = Z + dp 1 Z, then
d2K [] (r1 1)2:
Proof. Let R0 be the rectangle obtained by the application of Lemma 1.4.1
p
p
with v := 1 and w := d 1. Set := Sf1 + kd 1 : k 2 Zg. Clearly,
is a straight line cutting C in two half-planes. Denote by H the halfplane containing R0 and by H + the one containing R0 + 1. The rectangle R0
24
is fundamental, so, by (1.7), there are q0; q1 2 R0 such that (q0 ) = q0 and
(q1) = q1 + r1 : Observe that, because q0 + 1 = (q0 + 1) 2 (H +), we have
q0 = (q0 ) 2 (H ) and q0 + 1 = (q0 + 1) 62 (H );
and, because q1 + r1 + 1 = (q1 + 1) 2 (H +), also
q1 + r1 = (q1) 2 (H ) and q1 + r1 + 1 62 (H ):
Thus straight line segments in C, [q0 ; q0 + 1] and [q1 + r1 ; q1 + r1 + 1],
both intersect ( ) = @ ((H )). Consequently, if prx is the orthogonal projection on the real axis R C, then diam(prx( )) r1 1. This, using
-equivariance of , yields
l(1 ) diam(prx(1)) = diam(prx( )) r1 1:
Now we apply (1.9), (1.10) and (1.12) to get
K [] (B)=(B) l(1)2=d2 (r1 1)2=d2;
and we are done.
Q.E.D.
Fact 1.4.3 Suppose that e1 = 1, e2 = l cos + l sin p 1, l 2 (0; 1). If
sin2 (r1 1)2=(32K []);
then
K [] 41 (r1 1)1=3(r2 1)1=3 :
25
Proof. Put d := l sin . First assume that = =2. Fact 1.4.1, applied with
p 1), gives us
i = 1, j = 2 (i.e. ei = 1 and ej = d
8(1 + d2)K [] (r1 1)2 + (r2 1)2d2 (r2 1)2d2:
Inserting in the above inequality the estimate of d given by Fact 1.4.2, we
derive
8 2 K [] (r2 1)2(r1 1)2=K [];
and nally,
K [] 14 (r1 1)(r2 1):
(1:16)
In general, 6= =2 and then we consider := L L 1 , where
p
p
L : C ! C is given by L (x + y 1) := x + cot y + y 1 and transforms
p
p
the lattice Z + d 1Z to Z + (l cos + l sin 1)Z. The map L is areapreserving, so K [L ] equals the largest eigenvalue of L L , where L is the
adjoint of L . The trace of L L is 2 + cot2 , so we have
K [L ] 2 + cot2 :
Observe that (1.16) holds once is replaced by . Since by sub-multiplicativity
of q.c. dilatation
K [] K []=K [L ]2;
we get
K [] 14 (r1 1)(r2 1)=(2 + cot2 )2 :
(1:17)
Our hypothesis on can be written as
cot2 sin 2 32K []=(r1 1)2:
(1:18)
26
If cot2 1, then, by (1.17) and using r1; r2 6, we see that
2=3 2=3
K [] 4 132 (r1 1)(r2 1) 5 4 59 (r1 1)1=3(r2 1)1=3
14 (r1 1)1=3 (r2 1)1=3 :
If cot2 1, then, by (1.17) and (1.18), we obtain
K [] 14 (r1 1)(r2 1)=(3 cot2 )2 4 321 322 (r1 1)5(r2 1)=K []2;
which yields
K [] !
(r1 1)4 1=3 (r 1)1=3 (r 1)1=3
1
2
4 32 322
!1=3
54
1=3
1=3
1=3
1=3 1
(
r
1 1) (r2 1) (r1 1) (r2 1) :
2
2
4 3 32
4
Q.E.D.
Conclusion of the proof of Claim 1.4.1. We may assume that je1 j je2 j,
because otherwise one can switch the two. With the aid of a linear conformal
p
map we can further adjust e1 , e2 to e1 := 1 and e2 := l cos + l sin 1,
where l 2 (0; 1]. For \large" , namely when
sin2 (r1 1)2=(32K []);
we are done by Fact 1.4.3. Assume then that is \small", that is
sin2 (r1 1)2=(32K []):
Apply rst Fact 1.4.1 with i = 1 and j = 2 to see that
16K [] 8(1 + sin2 l2 )K [] (r1 1)2;
27
and so sin2 1=2. Thus, using r1 6, we verify that
q
q
r1 je1jj cos j je2j 6 1 sin2 l 6 1=2 1 > 0;
and application of Fact 1.4.1, with i = 2, j = 1, yields
8(l2 + sin2 )K [] (r2 1)2l2 + (r1 1)2 sin2 + (r1j cos j l)2
(r2 1)2l2 + (r1 1)2 2r1j cos j
(r2 1)2l2 + (r1 1)2=2;
(1.19)
where for the last inequality we used r1 6 again. Thus, dropping (r2 1)2l2
and isolating l2 , we get
l2 (r1 1)2=(2 8K []) sin2 (r1 1)2=(32K []) sin2 ;
and, applying (1.19) again, we derive
K [ ] (r2 1)2l2 =(16l2) + (r1 1)2=(32l2)
(r2 1)2=16 + (r1 1)2=32
1 (r 1)(r 1)
2
16 1
1 (r 1)1=3(r 1)1=3:
2
4 1
Q.E.D.
Chapter 2
Topological entropy at the asymptotic
average
Our ultimate goal in this work is to discuss topological entropy relative to
a vector in the rotation set. A large part of our discussion naturally ts in a
more general context, so we stray from the main course in this chapter. In the
rst section we dene topological entropy at the asymptotic average for a map
of a compact metrizable space with a continuous observable. We prove the
corresponding variational principle and use it for a rather weak version of the
thermodynamical formalism, which holds in this generality. Section 2.2 deals
with subshifts of nite type. Invoking the standard transfer matrix trick, we
give a real-analytic formula for the topological entropy at the asymptotic average. More importantly, we develop a technique that enables us to construct,
for any given asymptotic average, compact invariant sets \almost realizing"
the topological entropy at this asymptotic average.
28
29
2.1 Relative variational principle and thermodynamical formalism
Consider a compact metrizable topological space X with metric d and a
dynamical system generated by a homeomorphism f : X ! X . Moreover,
assume that a continuous function (called an observable ) : X ! Rd is
given. Our primary example will be a torus homeomorphism isotopic to the
identity with the displacement function of one of its lifts as the observable.
The rotation set becomes then a specialization of the set of asymptotic averages
(f; ) dened by
(f; ) :=
\
m>0
cl
[
n>m
!
f(1=n)Sn(f; )(x) : x 2 X g Rd;
(2:1)
where Sn(f; ) := (x) + ::: + (f n 1(x)) is the Birkho sum (cf. [Zie95]).
The set (f; ) is compact. Abstracting the proof of connectedness of the
rotation set in [MZ89], one can see that (f; ) is connected for connected X .
The complexity of the dynamics detected by an asymptotic average in
(f; ) may be measured by the associated topological entropy. In dening
this entropy one may follow a few paths leading to dierent invariants which
nevertheless coincide for the important examples of our primary concern (e.g.
for pseudo-Anosov homeomorphisms). Let us start with the most elementary
approach building directly on Bowen's denition of topological entropy (see
[Wal82]).
Given S X , we say that S is (; n)-separated if, for every two dierent
x; y 2 S , we have d(f k (x); f k (y)) , for some k 2 f0; 1; :::; n 1g. For a set
30
E Rd and > 0, we dene
sE (f; ; ; n) := maxf#S : S X (; n)-separated, 8x2S (1=n)Sn(f; )(x) 2 E g;
and, under the convention that ln 0 = 1, we set
hE (f; ; ) := lim
sup n1 ln sE (f; ; ; n) 2 f 1g [ [0; 1]:
n!1
The topological entropy associated with E is then
hEtop (f; ) := lim
hE (f; ; ):
!0
Notice that hE (f; ; ), as a function of , is decreasing, so the limit actually exists. Also, as suggested by the notation, hEtop (f; ) is a topological
invariant; it does not depend on the metric, nor is altered by replacing the
pair (f; ) with a (h 1 f h; h), where h is a homeomorphism of X (cf.
Proposition 2.1.2). Actually, one can easily make hEtop manifestly topologically
invariant by rephrasing its denition in terms of open covers, as it is done for
the usual topological entropy (see [Wal82]).
Also, hEtop (f; ; ) is monotonic in E and, for two sets E; F 2 Rd, we have
n
o
E [F (f; ) = max hE (f; ); hF (f; ) :
htop
top
top
(2:2)
Thus a bulk of information carried by hEtop (f; ; ) is still recoverable from
topological entropy at the asymptotic average, which is dened as the following
function of v 2 Rd
(v)
htop
(f; ) := rlim
hBr (v) (f; );
!0 top
where Br (v) is the open ball of radius r centered at v.
31
Actually, if we regularize E 7! hE (f; ) by setting
(E )
htop
(f; ) := rlim
hBr (E)(f; );
!0 top
(v)
(E )
(v)
then v 7! htop
(f; ) is characterized by the relation htop
(f; ) = maxfhtop
(f; ) :
v 2 E g, where E is a closed subset of Rd (cf. Lemma A.0.1).
(v)
The following is a summary of basic properties of htop
(f; ) that can be
easily derived from the denition | see also the appendix. In fact, part (ii)
(v)
below expresses the compatibility of htop
(f; ) and (f; ) that was one of the
(v)
axioms lying at the origin of htop
(f; ).
Proposition 2.1.1 (basics) With the above denitions, we have
(v)
(i) 1 htop
(f; ) htop (f ), v 2 Rd;
(v)
(ii) (f; ) = fv 2 Rd : htop
(f; ) 0g;
(v )
(iii) there exists v 2 Rd such that htop
(f; ) = htop (f );
(v)
(iv) htop
(f; ) is upper semi-continuous in v, i.e.
(v)
(v)
lim sup htop
(f; ) = htop
(f; ):
x!v
Proof of Proposition 2.1.1. Parts (i), (ii), (iv) require no comment. To see
(iii), note that (2.2) implies that, for any nite cover of (f; ), there exists
an element E of the cover with hEtop (f; ) = htop (f ). To nail down v simply
consider ner and ner covers (cf. Lemma A.0.1).
Q.E.D.
(v)
There are many properties of htop
(f; ) that are directly analogous to
those of the topological entropy, and the proofs are typically just simple modications of the standard ones. We mention here only those properties that
we will need in the further development.
32
Proposition 2.1.2 (Quotient Rule) Let (X; d) and (Y; e) be compact metric spaces and : X ! Rd . If f : X ! X is a factor of g : Y ! Y via
h : Y ! X (i.e. h is continuous surjective and f h = h g) then, for any
(v)
(v)
E Rd, hEtop (f; ) hEtop (g; h). Consequently, htop
(f; ) htop
(g; h),
v 2 Rd .
Proof. Fix an arbitrary > 0. By uniform continuity of h, there is > 0
such that e(y; y0) < implies d(h(y); h(y0)) < for all y; y0 2 Y . Now, given
S X which is (; n)-separated with Sn(f; )(x) 2 E for x 2 S , form S 0 Y
by choosing a point from h 1 (x) for each x 2 S . This set is (; n)-separated
for g and, for y 2 S 0, Sn(g; h)(y) = Sn(f; )(h(y)) 2 E . Since #S 0 = #S ,
we get hE (f; ; ) hE (g; h; ) hEtop (g; h), which nishes the proof
because of arbitrariness of .
Q.E.D.
Proposition 2.1.3 (Power Rule) For any m 2 N and E Rd, we have
hEtop (f m; (1=m)Sm()) = m hEtop (f; ):
In particular, for v 2 Rd ,
(v) m
(v)
htop
(f ; (1=m)Sm()) = m htop
(f; ):
Proof. This is a straightforward upgrade of the corresponding argument in
[Wal82]. We trace it for the sake of completeness. Fix E Rd. We will prove
that m hEtop (f; ) = hEtop (f m; (1=m)Sm(f; )). If S is an (; n)-separated set
for f m, then it is (; nm)-separated for f . Moreover,
(1=n)Sn(f m; (1=m)Sm(f; ))(x) = (1=(mn))Smn (f; )(x);
(2:3)
33
for any x 2 X . It follows that hEtop (f m ; (1=m)Sm()) m hEtop (f; ). On
the other hand, for > 0 there is > 0 such that d(x; y) forces
d(f i(x); f i(y)) < , for 0 i m. Let S be a (; n)-separated set for f m maximal among those that satisfy (1=n)Sn(f m; (1=m)Sm(f; ))(x) 2 E for all x 2
S . By the maximality and (2.3), for any y 2 X with (1=(mn))Smn (f; )(y) 2
E , there is x 2 S so that d(f km(x); f km(y)) < , 0 k n 1, and so
d(f i(x); f i(y)) < , 0 i mn 1. Consequently, if S 0 is an arbitrary
(2; mn)-separated set such that (1=(mn))Smn(f; )(x) 2 E , x 2 S 0, then it has
at most #S elements. By arbitrariness of S 0, one gets hE (f m; (1=m)Sm(); ) m hE (f; ; 2). It follows that hEtop (f m ; (1=m)Sm()) m hEtop (f; ). Q.E.D.
The next theorem, in the spirit of thermodynamical formalism ([Rue78]),
(v)
is a powerful tool for calculating htop
(f; ). The central role is played by the
restriction of the topological pressure functional P ([Wal82]) to the linear space
of functions fht; ()igt2Rd , where h; i is a scalar product in Rd. Namely, for
every t 2 Rd, we consider
t) (f; ) :=
p(top
lim
lim sup (1=n) ln max fPx2S exphSn(f; )(x); ti : S (; n)-separatedg :
!0
n!1
(t)
Note that p(0)
top (f; ) = htop (f ) and that htop (f ) = +1 implies ptop (f; ) = +1
t) (f; ) < +1 for all t 2 Rd ;
for all t 2 Rd. However, if htop (f ) < +1, then p(top
this is the situation we are mainly interested in.
t) (f; ) is a convex function of t. ConSince P is a convex functional, p(top
vexity will play a signicant role in our considerations, so let us x some
34
denitions before we go further. Our main reference is [Roc72]. For a convex
subset D Rd, denote by a(D) its ane hull, i.e. the smallest ane subspace
of Rd containing it. The interior of D with respect to the natural topology on
a(D) will be called relative interior and denoted ri(D). If g : Rd ! ( 1; +1]
is an arbitrary function, epi(g) := f(x; y) : y g(x)g is the epigraph of g, and
the set dom(g) := fx 2 Rd : g(x) < +1g is called the eective domain of g.
The closure of g is the function cl(g) dened by cl(g)(y) := lim infx!y g(x);
it is the greatest lower semi-continuous function majorized by g, and the
epigraph of cl(g) is the closure of the epigraph of g. The convex conjugate
g^ : Rd ! [ 1; +1] is dened by g^(y) := supfhx; yi g(x) : x 2 Rdg.
The convex conjugate is a lower semi-continuous convex function. If g is
a convex function, we have g^(y) = supfhx; yi g(x) : x 2 ri(dom(g))g
(see [Roc72] p.104). Convex conjugacy has an involutive property; namely,
(^g)^ = cl(convex hull(g)), where convex hull (g) is the pointwise supremum of
all convex functions bounding g from below. The epigraph of convex hull(g)
does not have to be equal to the convex hull of that of g. However, this is the
case when g is lower semi-continuous and has bounded eective domain (see
Proposition A.0.3 and its proof). Then also h := convex hull(g) is lower semicontinuous and h coincides with g at the extremal points of the epigraph of h.
In particular, g(x) = h(x) at the ordinates x of those extremal points that lie
over ri(dom(h)) | those values of x are referred to as points of strict convexity
of h. Finally, if we assume that htop (f ) < 1, then all the convex functions in
the following discussion are proper, i.e. they take values in ( 1; 1].
35
Theorem 2.1.1 (thermodynamical formalism) If htop (f ) < 1, then the
(v)
(v)
convex hull of v 7! htop
(f; ) is the convex conjugate v 7! p^top
(f; ) of t 7!
t) (f; ). In particular,
p(top
t) (f; ) ;
(v)
htop
(f; ) supd ht; vi p(top
t2R
and the equality holds at all points v where the right side is strictly convex.
Using (ii) of Proposition 2.1.1, we get the following straightforward corollary.
Corollary 2.1.1 If htop (f ) < 1, then the convex hull of the set of asymptotic
(v)
averages (f; ) is equal to the essential domain dom(^ptop
(f; )) of the convex
t) (f; ).
conjugate of p(top
Theorem 2.1.1 may be viewed as a large deviations type result (see [Ell85]).
It was inspired by Theorem II 6.1 in [Ell85]. In the appendix we give a broader
exposition together with an elementary proof. In this chapter we show Theo(v)
rem 2.1.1 as a byproduct of connecting htop
(f; ) with the main body of ergodic
theory. The major link in the connection is the following improvement of the
variational principle for topological entropy ([Wal82]).
Denote by M(X; f ) the set of all Borel probability measures invariant
under f and by E (X; f ) the set of all those measures in M(X; f ) that are
ergodic.
Theorem 2.1.2 (Relative Variational Principle) We have:
(i) for any closed convex set V Rd ,
Z
hVtop (f; ) sup h (f ) : 2 M(X; f ); d 2 V ;
36
(ii) for any open set U Rd ,
hUtop (f; ) sup
Z
h (f ) : 2 E (X; f ); d 2 U :
Thus, if we dene (using the convention that sup ; = 1)
and
Z
hverg (f; ) := sup h(f ) : 2 E (X; f ); d = v ;
(2:4)
Z
hvmeas(f; ) := sup h(f ) : 2 M(X; f ); d = v ;
(2:5)
then we have the following corollary.
Corollary 2.1.2 For v 2 Rd, we have
(v)
(f; ) lim sup hxmeas(f; ):
lim sup hxerg (f; ) htop
x!v
x!v
In particular,
cl ( hvmeas(f; )) := lim sup hxmeas(f; ) = convex hull
x!v
(v)
htop
(f; ) :
Proof of Corollary 2.1.2. The inequalities follow immediately from the deni-
tions. To justify the second assertion, note that, since the metric entropy h(f )
is ane as the function of , hvmeas (f; ) is a convex function of v. From the
ergodic decomposition theorem, hvmeas(f; ) is the convex hull of hverg (f; ).
(v)
We are done because htop
(f; ) is pinched between the two.
Q.E.D.
Also, by its convexity, hvmeas(f; ) is continuous at all points v in the
relative interior of its eective domain, ri((f; )); so, for those points, the
rightmost \lim sup" in Corollary 2.1.2 is superuous. Another instance when
37
we do not need that \lim sup" is when h (f ) is upper semi-continuous in
because then hvmeas(f; ) is upper semi-continuous in v. This happens for
expansive f ([Wal82]) and also for any f that is C 1-smooth ([New89]).
The following example shows that one can indeed have lim supx!v hxerg (f; )
(v)
< htop
(f; ).
Example. For (xi )i2Z 2 f1; 2; 3; 4gZ, (xk ; :::; xl ) is a parity block if xk ; :::; xl
are all even or all odd. We call it maximal if xk 1; xl+1 have the parity opposite
to that of xi, i = k; :::; l. Consider a subshift (; ) where
= fx 2 f1; 2; 3; 4gZ : x has at most two maximal parity blocksg
and shifts to the left: (((xi)i2Z ))i := xi+1. The space is compact and
invariant under . Let the observable : ! R be given by (x) := ( 1)x .
One can easily verify that (; ) = [ 1; 1]. All ergodic invariant measures
are supported on the union of even := fx 2 : xi even for all ig and
odd := fx 2 : xi odd for all ig because any recurrent point is contained in
one of the two sets. Nevertheless, h(0)
top (; ) ln 2. Indeed, for any n 2 2N,
consider points of with maximal parity blocks of the form (:::; xn=2 1 ) and
(xn=2 ; :::). They all have vanishing Sn(; ), and there are 2 2n such points
with pairwise dierent blocks of the rst n symbols.
Perhaps a more interesting example is := fx 2 f1; 2; 3; 4gZ :
0
x has no two nite maximum parity blocks of equal lengthg:
We leave analysis of this example to the reader.
38
Corollary 2.1.3 If 0 is an ergodic equilibrium state for ht0 ; i, that is ([Rue78,
Bow75, Wal82])
D
(t0 )
ptop
(f; ) = h0 (f ) + t0 ;
R
Z
E
d0 ;
v ) (f; ) = hv (f; ) = hv (f; ) = h (f ).
then, for v0 := d0, we have h(top
maes
erg
0
0
0
0
In particular, if is an ergodic measure of maximal entropy, then v :=
R d satises h(v ) (f; ) = h (f ) realizing thus the maximum of h(v )(f; ).
top
top
top
Such a measure exists for C 1 maps due to the result by Newhouse ([New89]).
Proof of Theorem 2.1.1 from Theorem 2.1.2. From the variational principle
for topological pressure ([Wal82]) we have
t) (f; ) = sup
p(top
Z
h (f ) + h; ti d : 2 M(X; f ) =
sup fht; vi + hvmeas(f; ) : v 2 dom ( hvmeas(f; ))g :
t) (f; ) = ( hv (f; ))^, so, by the involutive property of \^" and by
Thus p(top
meas
Corollary 2.1.2, we have
t) (f; ))^ = ( hv (f; ))^^ = cl( hv (f; )) = convex hull h(v) (f; ) :
(p(top
top
meas
meas
(v)
(We skipped \cl" before convex hull htop
(f; ) because this function is already lower semi-continuous | see Proposition A.0.3.)
Q.E.D.
Proof of Corollary 2.1.3. In the previous proof we got
(v)
p^top
(f; ) = cl( hvmeas(f; )); v 2 Rd :
(2:6)
39
Thus, combining trivial inequalities, we can write
cl ( hvmaes(f; )) hvmaes(f; ) hverg (f; ) h (f ) =
0
0
0
n
D Z
E t)
t) (f; )o = p^(v) (f; ) =
t0 ; d0 p(top
(f; ) supd ht; vi p(top
top
0
0
t2R
cl ( hvmaes(f; )) :
0
(v)
In this way, cl ( hvmaes(f; )) = hverg (f; ). By Corollary 2.1.2, htop
(f; ) is
pinched between the two, which nishes the proof.
Q.E.D.
0
0
Remark. If p(topt) (f; ) is dierentiable at t0 , then Corollary 2.1.3 is true without the hypothesis that 0 is ergodic. In fact, almost all measures in the
R
R
ergodic decomposition of 0 satisfy then h (f ) = h (f ) and d = d0.
R
Indeed, ane functions : t 7! h (f ) + ht; d i are all majorized by
t) (f; ) and, integrated with respect to , yield which is tangent to
p(top
t) (f; ) at t . Since there is a unique tangent at t , almost all 's must be
p(top
0
0
0
0
equal to which implies our claim.
0
The following fact will be used numerous times ahead.
Proposition 2.1.4 (Quotient Rule) Let (X; d) and (Y; e) be compact metric spaces and : X ! Rd be continuous. If f : X ! X is a factor of
g : Y ! Y via h : Y ! X (i.e. h is continuous surjective and f h = h g)
then, hverg (f; ) hverg (g; h), v 2 Rd .
Proof. Let M(Y ), M(X ) be the spaces of the Borel probability measures on
Y and X respectively, both equipped with the weak -topology. The map h
induces the push forward operator on measures, h : M(Y ) ! M(X ). Since,
40
for 2 M(Y; g), hh() (f ) h (G), it is enough to prove that h maps E (Y; g)
surjectively onto E (X; f ). This can be derived (via ergodic decomposition)
from the lemma 8.3 in [Man87]. We give a more elementary argument due
to Milnor. Fix an arbitrary 2 E (X; f ). Choose measures n 2 M(X )
supported on nite sets and converging to . By surjectivity of h, each atom
of n has a preimage in Y ; and, by taking those preimages with the obvious
weights, one easily gets n 2 M(Y ) such that h(n) = n. Passing perhaps
to a subsequence, we can assume that n's converge to a measure 2 M(Y ).
Since h is continuous, h ( ) = . To make invariant under g, replace with any limit point of the sequence of time averages (1=n) Pin=11 gi ( ) n2N
| those averaged measures also map under h to . Thus we got 2 M(Y; g)
such that h ( ) = . This shows that a compact convex set C := h 1 () \
M(Y; g) is not empty; therefore, it has an extremal point 2 M(Y; g). The
measure must be ergodic. Indeed, if it were a nontrivial convex combination
of two other measures 1; 2 2 M(Y; g), then, by extremality of in C , h(i ) 6=
, i = 1; 2. This contradicts ergodicity of because = h ( ) is a convex
combination of h(1 ) and h(2 ).
Q.E.D.
Proof of Theorem 2.1.2. We build on Misiurewicz's proof of the variational
principle for topological entropy as found in [Wal82]. Basically, we need to
augment it with arguments that keep track of the averages of the observable.
This is trivial for the proof of (i). Then the natural measures (i.e. limits of
those equidistributed on the appropriate (; n)-separated sets) have the right
averages, and we just quote [Wal82] claiming that their entropy is as required.
41
Part (ii) is more involved; we are forced to interweave the proof from [Wal82]
with several arguments including application of Egoro's and Birkho's theorems.
Proof of (i). We will show that, for any > 0, there is 2 M(X; f ) with
R d 2 V and h (f ) > hV (f; ) . Choose > 0 so that hV (f; ; ) >
top
hVtop (f; ) . From the denition of hV (f; ; ), there exists a sequence nk 2 N,
nk ! 1 and (; nk )-separated sets Ek such that limk!1(1=nk ) ln #Ek =
hV (f; ; ) and (1=nk )Snk (f; )(x) 2 V , for x 2 Ek . Consider the uniform
k 1 f j.
atomic measures on Ek , call them Ek , and set k := (1=nk ) Pnj=0
Ek
R
R
Note that, by convexity and closedness of V , dk = (1=nk )Snk (f; ) dEk 2
V . Passing perhaps to the subsequence of (nk ), we can assure that k 's weak R
converge to some probability measure . Clearly d 2 V . More importantly, Misiurewicz proves that h (f ) limk!1 1=nk ln #Ek (see part (2) of
the proof of Th.8.6., p. 189 in [Wal82]), so h(f ) hV (f; ; ) > hVtop (f; ) .
Q.E.D.
Proof of (ii). We will show that: if hUtop (f; ) > 2 and 2 M(X; f ) is such
that
Z
nlim
!1(1=n)Sn (f; )(x) = := d 2 U;
(2:7)
for -almost all x in X , then h (f ) hUtop (f; ) + ln 2 + 1. Clearly, (2.7) is
R
satised by ergodic with d 2 U ; however, unlike ergodicity, (2.7) is
invariant under replacing f and by the iterate f m and by (1=m)Sm(f; )
respectively. By considering arbitrarily high iterates and using the power rule
(Proposition 2.1.3), we conclude that h (f ) hUtop (f; ).
42
Let A = A1; :::; Ak be an arbitrary nite Borel partition of X . Choose
> 0 so that < 1=(k ln k). Using Egoro's theorem, we may nd compact
sets Bj Aj , with (Aj nBj ) < , 1 j k, such that (1=n)Sn(f; ) converges
uniformly to on every Bj . Now, because of this uniform convergence and
openness of U and since 2 U , there is 0 < d < 1 such that
(1=m)Sm(f; )(x) + 2d f(x) : x 2 X g U;
(2:8)
for any x 2 Skj=1 Bj , and suciently large m 2 N.
Put B0 := X n SfBj : 1 j kg and introduce a partition B :=
fB0; B1 ; :::; Bk g. Now, for 1 j k, we have Bj Aj , so the entropy
H(A=Bj ) of A restricted to Bj vanishes for those j 's. Also, by the trivial
estimate (see Corollary 4.2.1 p. 80 in [Wal82]), H (A=B0) ln k. Thus we
get conditional entropy H(A=B) = P0jk (Bj )H (A=Bj ) (B0) ln k k ln k < 1. Consequently, (by (iv) of Th.4.12, p. 89 in [Wal82]) we have
h (f; A) h (f; B) + H (A=B) < h(f; B) + 1:
(2:9)
For each n, let us split Bn := Wjn=01 f j B into B1n := fB 2 Bn : B B0 \ ::: \ f bdnc B0g and B2n := Bn n B1n. Note that #B1n (#B)n bdnc , thus,
for suciently large n, we have
#B2n #Bn (#B)n bdnc (1=2)#Bn:
(2:10)
Put Kn := fx 2 X : (1=m)Sm(f; )(x) 2 U for all m ng. We claim
that the union S B2n is contained in Kn for suciently large n. Indeed, if
x 2 A 2 B2n, then there is s, 0 s bdnc; such that f s(x) 2 Bj with j 6= 0.
43
Now, a simple estimate gives
k(1=m)Sm(f; )(x) (1=m)Sm(f; )(f s(x))k
2bdnc sup
k(z)k=m 2d zsup
k(z)k;
z2X
2X
so, by the choice of d (see (2.8)), we have (1=m)Sm(f; )(x) 2 U .
Adopting Misiurewicz's trick, consider C := fB0 [ B1 ; :::; B0 [ Bk g. This is
an open cover of Kn (also X ). Write N (C n ; Kn) for the minimal cardinality of
a subfamily of C n := Wjn=01 f j C that covers Kn, and let Cn be such a subfamily
with #Cn = N (C n ; Kn). By our claim, each B 2 B2n lies in Kn; therefore, there
exists C 2 Cn containing B . It is also clear from the denition of C that at
most 2n elements of B2n can t into any xed element of Cn . We conclude that
#B2n N (C n; Kn)2n:
(2:11)
By Th. 7.7 in [Wal82], if is the Lebesgue number of C , then N (C n ; Kn) s(=2; n; Kn), (where s(=2; n; Kn) is the maximal cardinality of an (=2; n)separated subset of Kn). Note that every (=2; n)-separated subset of Kn has
the averages (1=n)Sn(f; )(x) in U , for large n. Thus, for those n, we get
N (C n ; Kn) sU (f; ; =2; n).
Now, combining (2.10), (2.11) and the last inequality, we get, for large n,
H(Bn) ln #Bn ln(2#B2n) ln 2n+1sU (f; ; =2; n) :
Passing to the limit, rst with n ! 1 then with ! 0, we obtain
h (f; B) ln 2 + hUtop (f; ):
44
From (2:9), h (f; A) ln 2 + hUtop (f ) + 1, and by arbitrariness of A we
get h(f ) ln 2 + hUtop (f; ) + 1, as promised.
Q.E.D.
2.2 Topological entropy at the asymptotic average for subshifts of nite type
(v)
As an example of utilizing Theorem 2.1.1 to calculate htop
(f; ), we consider the case of a transitive subshift of nite type with locally constant observable. This setting is exactly what we will need to analyze the case of
pseudo-Anosov maps in the next chapter.
t) (f; )
In the rst subsection, using standard techniques, we identify p(top
with the leading positive eigenvalue of the appropriate transfer matrix. This
(v)
(v)
gives an explicit real analytic formula for htop
(f; ). We also note that htop
(f; )
is realized as the metric entropy for the corresponding Gibbs state.
The second subsection is devoted to showing that, apart from having the
associated ergodic Gibbs state, each v 2 ri((f; )) is the asymptotic average
for a compact invariant set with the topological entropy arbitrarily close to
(v)
htop
(f; ).
In most of our formulations we restrict attention to v in ri((; )) since
ultimately (in the case of torus homeomorphisms) only such rotation vectors
will be considered. However, the case of v in the boundary of (f; ) can be
reduced to that of v 2 ri((; )) by passing to a smaller subshift of nite type
using the scaold construction (taken from [MT91]). We briey describe this
45
(v)
trick in the last subsection and show how it relates to htop
(f; ).
(v)
For subshifts of nite type, the entropy htop
(f; ) contains much more
rened information that what is suggested by its denition. It can actually
be used to count the number of periodic orbits with a prescribed average of
the observable ([MT91]). This belongs to a topic that parallels the classical
theorems on the distribution of the prime numbers via the formalism of Zeta
functions. There is a large amount of published material on the subject, see
for example [PP90] and the references within.
2.2.1 Transfer matrix, Gibbs states, and explicit formulas
Consider a transitive subshift of nite type : ! with a locally constant observable function : ! Rd. Without any loss of generality, we can
assume that there is an irreducible ([Wal82]) binary matrix A = (ai;j )1i;jN
and a Rd-valued matrix ( i;j )1i;jN such that = fx 2 f1; :::; N gZ : axi ;xi =
1; i 2 Zg and (x) = x ;x for x 2 . The metric d on , that we shall use,
is given by d((xk ); (yk )) := 2 jkj, where k is the closest to 0th position at which
xk and yk dier.
+1
0
1
A convenient representation of this setting is achieved by interpreting
points of as bi-innite paths in the transition graph G f1; :::; N g2 with
vertices f1; :::; N g and an edge (i; j ) from i to j whenever ai;j = 1. By a path
in G we understand a map : fk; :::; l 1g ! G, where k; l 2 Z[f 1; +1g,
and (i + 1) emanates from the endpoint vertex of (i), k i < l 1. The
46
number l
k is the length of the path, denoted by l(). A loop is a path
with the endpoint of (l 1) coinciding with the starting point of (k). The
loops visiting any vertex of G at most once are called elementary. Clearly,
there is a nite number of the elementary loops in G. Each periodic orbit of
determines uniquely a loop with the length equal to the period. To keep
track of the averages of , each edge (i; j ) 2 G is tagged with the vector weight
P
i;j . By we denote the sum of the weights along the edges of the path
. If is a loop, we write () for the average (1=l()) P . Transitivity of
guarantees that any pair of vertices i; j 2 G can be connected by a path i;j
in G. We will refer to the paths in a xed collection fi;j g(i;j)2G as connecting
paths and call t := maxfl(i;j ) : i; j g a connecting length of G.
The shift map is expansive with the expansiveness constant = 1 (see
[Wal82]); therefore, in an analogous way to the usual topological entropy, one
(v)
can easily verify that htop
(; ) can be calculated on (1; n)-separated sets. If
S is (1; n)-separated, then it determines a collections of #S dierent
paths in G of length n | for a point (xi ) 2 S , the corresponding path passes through x0 ; :::; xn and Sn(; )(x) = P . Clearly, every collection of
dierent paths of length n arises from a (1; n)-separated set in this way. Thus
we can write
(v)
htop
(; ) = rlim
lim sup(1=n) ln sBr (v) (; ; 1; n) =
!0 n!1
lim lim sup(1=n) ln #f : path in G; l() = n; (1=n)
r!0
n!1
X
(2:12)
2 Br (v)g:
Since every path can be extended to a loop by adding an appropriate
47
connecting path, we have
(v)
htop
(; ) = rlim
lim sup(1=n) ln #f : loop in G; l() = n; () 2 Br (v)g:
!0 n!1
(2:13)
One can even insist that all the loops above depart from an arbitrarily xed
vertex. The \lim sup" in (2.13) can not be replaced with \lim inf" unless is
mixing. This (slight) inconvenience is oset by the following simple fact (cf.
the example in Appendix A).
Fact 2.2.1 In the formula (2.12), one can replace lim sup with lim inf.
Proof. Take an arbitrary 0 < < 1. We will show that with \lim inf" we get
(v)
at least htop
(; ). Fix v 2 Rd and r > 0. By (2.13), there exist arbitrarily
large n0 2 N such that there is a collection of dierent loops of length n0
(v)
with cardinality # exp (htop
(; )n0 ) and (1=n0) P 2 Br=2(v), 2 .
As noted above, we can assume that all these loops depart from a xed vertex
of G so that we can concatenate them. For any n 2 N, set k := bn=n0 c. By
considering all possible (# )k concatenations of k-tuples from k , we conclude
that
(v)
sBr (v) (; ; 1; n) sBr= (v) (; ; 1; kn0 ) (# )k exp (htop
(; )kn0 )
2
(where for the rst inequality n must be suciently large). Taking the logarithms and \lim inf" with n ! 1 nishes the proof.
Q.E.D.
Combination of the results in the preceding section of this chapter with
the known facts about subshifts of nite type gives us the following theorem
48
| very much in the spirit of [Bow75, Rue78]. (A reader not familiar with the
Gibbs states may want to consult [Bow75].)
Theorem 2.2.1 Suppose that (as above) : ! is a transitive subshift
of nite type with observable : ! Rd that depends only on the rst two
symbols. If A = (ai;j )1i;jN is the corresponding binary matrix and G is the
corresponding transition graph with weights (
i;j )1i;j N ,
then:
(i) (; ) is a convex polyhedron with vertices in
f() : is an elementary loop in Gg;
(v)
(ii) htop
(; ) is convex continuous on (; ) and strictly convex and real
analytic on ri((; ));
t) (; ) = ln((t)), where (t) is the leading positive eigenvalue of the
(iii) p(top
transfer matrix A(t) = (ai;j expht; i;j i)1i;j N , t 2 Rd ;
(v)
(v)
(iv) htop
(; ) = p^top
(; ), for v 2 ri((; ));
(v)
(v) htop
(; ) = hverg (; ) = hvmeas(; ) = h v (), where v 2 ri((; ))
( )
and (v) is the Gibbs measure corresponding to the function on given by
t) (; ) (where r
x 7! ht; (x)i, with t 2 Rd uniquely determined by v = rt p(top
t
is the gradient with respect to t).
Remark. The dimension of (; ) may be lower than d. Thus real analyticity
in (ii) is with respect to the ane hull of (; ). Nevertheless, even though
we have chosen not to do so, one can always assume that (; ) has the
dimension of the target space of . This is achieved by replacing with its
projection onto the ane hull of (; ).
49
Remark. Let us add to (v) that, since is locally constant, the Gibbs measure
is a Markov measure. More specically, if u = (ui)i=1;:::N and v = (vi)i=1;:::N
are the eigenvectors, corresponding to the eigenvalue (t), of A(t) and of the
transpose A(t) respectively, then this Markov measure is determined by a
unique stochastic vector proportional to (uivi )i=1;:::N and by the stochastic
matrix ( 1A(t)i;j vj =vi)i;j=1;:::N (cf. p. 194 in [Wal82].) Also, ui; vi > 0,
i = 1; :::; N , so this measure is supported on the whole .
Remark. For a subshift of nite type that is not transitive, the entropy
(v)
htop
(; ) is the maximum of the entropies on the transitive components.
The terminology transfer matrix comes from statistical mechanics (see
[Ell85, Rue78]). Part (iii) of Theorem 2.2.1 is the essence of the transfer
matrix method (see [May91] for an exposition).
Proof of (i). This has been apparently realized by many; it is implicitly con-
tained in [Fri82b], mentioned in [Kwa92], and proved in the full generality in
[Zie95]. One way to argue is as follows. By transitivity, the rotation set is the
closure of the -averages ( ) taken over all loops in G. Thus it is enough
to see that, for every loop , the sum P can be represented as a sum of
P 's for a number of elementary loops . This can be done by induction
with respect to the length of . If is not elementary, then certain vertex
v repeats at least twice and, by taking the two segments of between two
dierent occurrences of v, one can \pinch" to two shorter loops without
changing the sum of weights.
Q.E.D.
50
Proof of (iii). The topological pressure is given by
t) (;
p(top
) = nlim
!1(1=n) ln
X(
)!
D X E
exp t;
: path in G; l() = n ;
(2:14)
or, in terms of loops, we can write (cf. (2.13) and [May91])
t) (;
p(top
) = lim sup(1=n) ln
n!1
X(
)!
D X E
exp t;
: loop in G; l() = n :
(The convergence takes place for mixing .) The right-hand side can be written
as
lim sup(1=n) ln(trace(A(t)n)) = ln((t));
n!1
where the matrix A(t) = (ai;j expht; i;j i)1i;jN , and (t) is its leading positive
eigenvalue given by the Perron-Frobenius theorem ([Wal82]).
Q.E.D.
Proof of (iv). Because A(t) is a non-negative irreducible matrix, the lead-
ing positive eigenvalue (t) is a simple root of the corresponding characteristic polynomial, and so it depends real-analytically on t 2 Rd. In this way
t) (; ) is smooth; and consequently, its convex conjugate is strictly convex
p(top
on the interior of the eective domain (see p. 253, [Roc72], cf. (2.15) below).
We are done by Theorem 2.1.1. ( The eective domain is equal to (; ) by
Corollary 2.1.1.)
Q.E.D.
t) (; ) is smooth, its convex conjugate p^(v) (; )
Proof of (v). Since by (iii) p(top
top
(v)
can be calculated for v 2 ri dom p^top
(; )
[Roc72])
as follows (cf. Th. 23.5,
(v)
t) (; );
p^top
(; )) = hv; ti p(top
(2:15)
51
(t)
where t is uniquely determined by rt ptop
(; ) = v. Moreover, we observed
(v)
while proving (iv) that (; ) = dom(^ptop
(; )).
Let (v) be the Gibbs state for ht; i. According to Ruelle (see Th.1.22 in
[Bow75]) it is an (ergodic) equilibrium state, that is
t) (;
p(top
E
D Z
) = h v () + t; d(v) :
( )
The ane function : t0 7! h v () + ht0;
( )
R d(v) i is majorized by the convex
t) (; ) due to the variational principle, and, by the above equality,
function p(top
t) (; ). Comparison with (2.15) yields R d(v) = v
it is actually tangent to p(top
(v)
and h v () = p^top
(; ). We are done by (iv) and Corollary 2.1.3. Q.E.D.
( )
(v)
Proof of (ii). While proving (iv) we observed strict convexity of p^top
(; ) =
(v)
t) (; ),
htop
(; ), for v 2 ri((; )). By (2.15) and real-analyticity of p(top
(v)
(; ) is real-analytic on ri((; )).
v 7! htop
Still, one needs to show convexity on the whole (; ). Then continuity
(v) (; ) is lower semi-continuous by (iv) of Proposition
follows; indeed, htop
2.1.1, and, by Th. 10.2, p.84 [Roc72], it is upper semi-continuous as a convex
function on a polyhedron.
To demonstrate convexity, it is enough to show the following: if v1 ; v2 2
(; ) and v = v1=2 + v2 =2, then
(v) (; ) (1=2)h(v ) (; ) + (1=2)h(v ) (; ):
htop
top
top
1
2
It is routine to extend this inequality to 2-adic convex combinations and then
(v)
(by upper semi-continuity of htop
(; )) to arbitrary convex combinations.
52
Fix an arbitrary r > 0. By (2.12) and Fact 2.2.1, there are arbitrarily
large n0 2 N for which one can nd two families 1 and 2 of paths 1 in G,
(vi )
with length n0, so that # i exp (htop
(; ) r)n0 and P 2 n0 Br (vi)
for 2 i, i = 1; 2. Moreover, we may assume that n0r exceeds the connecting
length t.
For each pair of paths (1; 2) in 1 2 we can nd a connecting path so that := 12 is a path in G. At least # 1# 2 =t of thus obtained paths
will have the same length
n := l( ) = 2n0 + l (1 + r)2n0;
where l := l( ). Let
then
(2:16)
be a collection of those 's. Observe that, if 2 ,
(1=n)
X
2 Br+ (v);
where accounts for the contribution introduced by and is smaller than r
for large n0 . Thus we have
v)
v)
sB r (; ; 1; n) # exp (h(top
(; ) r)n0 exp (h(top
(; ) r)n0 =t:
2
1
2
By passing with n0 to innity, while keeping in mind (2.16), we get
hBtopr (v) (; ) (2(1 + r))
2
1
(v )
v ) (; ) 2r :
htop (; ) + h(top
1
2
We are done by the arbitrariness of r.
1 These
can be actually chosen to be all loops through a xed vertex.
Q.E.D.
53
2.2.2 Observable shadowing with the right entropy
Let , , , and G be as in the preceding subsection. By (v) of Theorem
2.2.1, we know that, for any v in ri((; )), there is an ergodic measure with
(v)
the metric entropy equal to htop
(; ). That measure is supported on all of (see the remark after Theorem 2.2.1), and the support could not possibly be
smaller as shown by the following proposition.
Proposition 2.2.1 Suppose that v 2 ri((; )). If K is a compact -invariant
(v)
(v)
(jK ) < htop
(; ).
proper subset of with ( jK ; ) = v, then htop
The proposition also demonstrates that the following theorem is optimal.
Theorem 2.2.2 For v 2 ri((; )) and 0 < < 1, there exists a compact
(v)
-invariant set K such that htop (jK ) htop
(; ) and (jK ; ) = v.
In fact, K is such that, for some constant C , kSn(; )(x) nv k C , x 2 K ,
n 2 N.
This theorem, for the special case of with values in Z and v 2 Z, is
proved in [MT92]. Also, in [Zie95] it is shown how to nd for each v 2 (; )
a compact invariant set K with (jK ; ) = v. Our methods, developed
independently of [MT92, Zie95], are dierent. Let us also note that, using the
scaold construction of the next subsection, one can get a stronger result for
hverg , see Theorem 2.2.4.
The proposition is simple, so we prove it rst. We need the following
lemma.
54
Lemma 2.2.1 If K is a compact, -invariant proper subset of , then, for
(t)
t) (; ).
any t 2 Rd , we have ptop
(jK ; ) < p(top
Proof of Proposition 2.2.1 from Lemma 2.2.1. Pick v 2 ri((; )). There is
(v)
(t )
the corresponding t0 2 Rd such that htop
(; ) = hv; t0i ptop
(; ), (see
0
(2.15)). Using Theorem 2.1.1 for the rst inequality and the lemma for the
third, we obtain
(v)
t) ( j ; )g htop
(jK ; ) sup fhv; ti p(top
K
t2R
(t )
(v)
(t )
hv; t0i ptop
(jK ; ) > hv; t0i ptop
(; ) = htop
(; ):
2
0
0
Q.E.D.
Proof of Lemma 2.2.1. Let be the Gibbs measure associated with ht; i. Its
characteristic property is that there is a constant C > 0 such that, for any
x = (xi)i2Z 2 and n 2 N, ([Bow75])
t) (; ) C;
C 1 ([x]n0 )= exp hSn(; )(x); ti np(top
(2:17)
where [x]n0 := fy 2 : yi = xi ; i = 0; :::; n 1g is a n-cylinder. Since K is
proper and is ergodic and supported on the whole , we have (K ) = 0. We
can contain K in a union of disjoint cylinders that total to arbitrarily small measure; more concretely, for suciently high n0 2 N, there exists a discrete
subset En of such that K Sx2En [x]n0 and (Sx2En [x]n0 ) (10C ) 1.
Now, for n 2 n0 N, consider any (1; n)-separated subset of S K . Any n0cylinder that is determined by n0 consecutive symbols of a point in S must be
0
0
0
0
0
55
one of [x]n0 ; x 2 En . This, together with (2.17), yields the following estimate
0
0
X
n=n
Y0
X
i=1 x2En0
x2S
exphSn(; )(x); ti n=n
Y0
X
i=1 x2En0
exphSn (; )(x); ti 0
t)
t)
n=n
C([x]n0 ) exp np(top
(; ) C (10C ) 1 exp n0p(top
(; )
:
0
0
Taking the logarithms and passing to the limit with n ! 1, we get (by (2.14))
t) ( j ; ) p(t) (; ) + (1=n ) ln(1=10):
p(top
K
0
top
Q.E.D.
We turn our attention to Theorem 2.2.2 now. As most of proofs in this
section our argument hinges on concatenation of paths in G, however this
time we need an original approach to control the averages. Sacricing the
eciency of exposition, we take an opportunity to present a general technique
that easily yields trajectories in that have an a priori prescribed behavior of
the averages. (See Lemma 2.2.2, ahead.)
We start with a simple characterization of convex polyhedra that may be
interesting by itself. (Below, for two sets A; B Rd, A + B stands for the
Minkowski sum, i.e. A + B = fa + b : a 2 A; b 2 B g.)
Theorem 2.2.3 (characterization of convex polyhedra) For a compact
convex 0 Rd , the following are equivalent :
(i) 0 is a polyhedron;
S
(ii) there exist m 2 N and !1 ; :::; !m 2 0 such that 0 + 0 = mi=1 !i + 0 :
Moreover, the !i's can be chosen so that they are convex combinations with
rational coecients of the extremal points of 0 .
56
Proof of Theorem 2.2.3. To prove that (ii) implies (i), we just notice that, for
x; y 2 0 : if the sum x + y is an extremal point of 0 + 0 = 2 0 , then both x
and y must be extremal and equal to each other. Thus m bounds from above
the number of extremal points of 0 what makes it a polyhedron.
To prove (ii) from (i), we rst note that ane transformations of 0 do not
aect validity of (ii). Since every polygon with n vertices is an ane image
of the standard simplex n = fx 2 Rn : xi 0; P xi = 1g Rn, we may
consider only the case when 0 = n.
We claim that
[
2 n = fw + n : w = (w1; :::; wn) 2 n ; wi 2 (1=n)N; i = 1; :::; ng:
To get this, x any x = (x1 ; :::; xn) 2 2 n . Clearly Pni=1 xi = 2, so we
see that Pni=1bnxi c Pni=1(nxi 1) = 2n n = n: Hence, we can choose
ki 2 N, 0 ki bnxi c; i = 1; :::; n, such that Pni=1 ki = n. Now, by setting
wi := ki=n and yi := xi wi; i = 1; :::; n, we get Pni=1 wi = n=n = 1 and
Pn y = 2 1 = 1, so both w = (w ) and y = (y ) belong to n. Clearly
i
i
i=1 i
x = w + y.
Q.E.D.
The following \shadowing type result" is the consequence of Theorem
2.2.3 that we are interested in.
Corollary 2.2.1 Suppose that 0 and !i's are as in Theorem 2.2.3. If x :
R+ ! Rd is piecewise C 1 with x(0) = 0 and x0 (t) 2 0 , t 2 R+, then there
exists a sequence 2 f1; ::; mgN such that x(n) 2 ! + ::: + !n + 0 , for all
n 2 N.
1
1
57
Proof. We proceed by induction with respect to n. Since x0 (s) 2 0 , by
R
convexity of 0 , x(1) = 01 x0 (s)ds 2 0 taking care of the case n = 1. For
R
the induction step, write x(n + 1) = x(n) + nn+1 x0 (s)ds 2 x(n) + 0 and use
induction hypothesis to see that x(n +1) 2 ! + ::: + !n + 0 + 0 . By (ii) of
Theorem 2.2.3, one can to nd 1 n m so that x(n+1) 2 ! +:::+!n +0 .
Q.E.D.
1
1
1
Let us now explain how Corollary 2.2.1 renders orbits in (or equivalently
paths in the graph G) with prescribed behavior of the Birkho averages of .
By transitivity, we can nd loops 1, ... r , all emanating from a common
arbitrarily chosen vertex i0 , such that 0 := convf(1 ); :::; (r )g approximates
(; ) as well as we wish. Now, let !i, i = 1; :::; m, be as in (ii) of Theorem
2.2.3. Since !i's are all rational convex combinations of the vertices, by taking
appropriate concatenations of i's, we can obtain loops 1; :::; m so that !i :=
(i), i = 1; :::; m. Moreover, replacing each loop i with its appropriate
multiple we can make all i's of equal length l0. One can concatenate i's at
will: if q 2 N and 2 f1; :::; mgq , then := :::q is a loop in G.
1
The following lemma is in fact much stronger than what we need for
Theorem 2.2.2. For a path in G, by mn we understand its restriction to
fm; :::; n 1g (if it is well dened).
Lemma 2.2.2 If x : R ! Rd is piecewise C 1 with x(0) = 0 and x0(t) 2 0,
t 2 R, then there exists a bi-innite path in G following x, i.e.
X
n
m
(x(n) x(m)) A := 4l0 diam(0 ) + 2l0 sup k (x)k;
x2
58
for all n < m; n; m 2 Z. Moreover, for n 2 l0 N, the edge (n) ends at the
vertex i0 .
Clearly, the smoothness hypothesis is supercial.
Proof. By applying Corollary 2.2.1 to the rescaled path t 7! (1=l0 )x(l0 t) twice,
once with time running forward (t > 0), once with time running backward
(t < 0), we get 2 f1; :::; M gZ such that for any k; l 2 Z with k < l,
X
l 1 ! (1=l0)(x(l0 l) x(l0 k)) 2 diam(0 ):
i =k i
Take for the innite concatenation := = ::: ::: : By multiplying
the above inequality by l0, while remembering that !i = (i ) = (1=l0) Pi ,
we see that
1
0
1
(x(ll0 ) x(kl0 )) 2l0diam(0):
The above inequality for restrictions of to intervals between multiples of l0
yields the result because, for m; n with jm nj l0 , both k Pmn k and kx(n)
x(m)k are trivially bounded by l0 supx2 k (x)k and l0 diam(0) respectively.
X
kl
ll
0
0
Q.E.D.
Proof of Theorem 2.2.2. We will construct a sequence of natural numbers
ni ! 1 and a sequence of (1; ni)-separated sets Ei such that
and, for y 2 Ei,
(v)
#Ei exp(htop
(; )ni)
(2:18)
kSk (; )(y) kvk C;
(2:19)
59
where k 2 N and C is a constant independent on i. Then the set K :=
cl(Sl2N;i2N l (Ei)) satises the assertion of Theorem 2.2.2. This is because
(2.19) is satised on a closed set of y 2 and, if it holds for y, then it holds
for (y); 2(y); ::: with C replaced by 2C .
Fix v 2 ri((; )). In constructing Ei we will think of its points as
innite paths in G. As in the discussion preceding Lemma 2.2.2, choose loops
1 , ...m so that v 2 ri(0 ), where 0 = convf(1); :::; (m )g. Recall that all
i's have a common starting vertex i0 . Let > 0 be rational and such that
1=(1 + ) p. Take r > 0 satisfying v B2r= (0) 0 . From the denition
(v)
of htop
(; ) rephrased as (2.13), nd n0 2 N and a collection S0 of loops in
(v)
G, with length n0 and starting at i0, such that #S0 exp(htop
(; )pn0 )
and (1=n0) P 2 Br (v) for any 2 S0. Require also that n0 is large enough
so that
A = 4l0 diam(0) + 2l0 sup k (x)k rn0:
x2
(2:20)
With no loss of generality one can assume that n0 2 l0 N.
Set ni := in0 (1 + ), i = 1; 2; ::: .
Fix i 2 N, and choose 1; :::i 2 S0 arbitrarily. With this data we will
associate a path of length ni and of the form 1122::::ii , where j 's will
be chosen so that
X
1 1 :::j j
nj v rn0; j = 1; 2; :::; i:
(2:21)
60
P
We dene j 's inductively. We start with nding 1. Observe that, since
2 n0 v + Bn r (0),
1
0
n0 (1 + )v
X !.
(n0 ) 2 (n0 v + Bn r (0))=(n0) = v + Br= (0) 0:
0
1
Consequently, there is a (linear) path x(t) 2 Rd; 0 t n0 , such that
x0 (t) 2 0 ; t 2 [0; n0], x(0) = 0, x(n0 ) = n0(1 + )v P . From Lemma
2.2.2, we get a path 1 such that l(1) = n0 and
1
X
x(n0 ) A rn0 :
1
Note that 1 is in fact a loop that starts (and ends) at the vertex i0 , and so is 1;
thus we can concatenate the two. By the denition of x, P +x(n0 ) n1 v =
0, so from the above inequality we verify (2.21) for j = 1:
1
X
1 1
X
X
+ x(n0 ) n1 v + n1 v 1
1
x(n0 ) rn0: (2:22)
To get j+1, having 1; :::; j already dened and satisfying (2.21), we
repeat the above procedure with a tiny modication. Observe that, since
P
2 n0 v + Bn r (0), using (2.21) as the induction hypothesis, we get
j
+1
0
0
@nj+1v
1
.
A (n0 ) =
:::j j j
0
1
.
X
X
@n0 v + (n0 v
) + (nj v
)A (n0)
X
1 1 2
j+1
+1
1 1 2 :::j j
2 (n0v + B2n r (0))=(n0) = v B2r= (0) 0 :
0
As before, we conclude that there is a (linear) path x(t) 2 Rd; 0 t n0 ,
such that x0 (t) 2 0 ; t 2 [0; n0 ], x(0) = 0, x(n0 ) = nj+1v P :::j j .
1 1
+1
61
From Lemma 2.2.2, we get a path j+1 such that l(j+1) = n0 and
X
j+1
x(n0 ) A rn0:
For the concatenation 11:::j+1j+1 we have
nj+1v ::::j j
X
X
+ x(n0 ) nj+1v + j
::::j
1 1
X
1 1
+1
+1
+1
+1
x(n0 ) 0 + rn0:
This ends the induction step.
The above procedure yields (#S0)i dierent paths of the form =
11:::ii, because we have that many choices of 1; :::; i 2 S0. To get from
each such a point of Ei , just extend the path = 11:::ii to a bi-innite
path. In this way, Ei is indeed a (1; ni)-separated set with
i
(v)
(v)
(; )nip=(1 + ) :
#Ei = (#S0 )i exp(htop
(; )pn0) = exp htop
By the choice of , (2.18) follows. The (2.19), for k ni, is an immediate
consequence of (2.21) (with C depending on n0 ). To get it right for all k,
one just has to exercise some care while extending 's to the bi-innite paths.
Namely, the semi-innite paths that are concatenated to must follow x(t) :=
tv, as in Lemma 2.2.2.
Q.E.D.
2.2.3 The scaold of subshifts of nite type
The following discussion (based on [MT91]) explains how to use the re(v)
sults of the previous sections to learn about htop
(; ) and hverg (; ) even if
62
v 62 ri((; )). Since (; ) is a convex polyhedron, we can think of (; )
as the total space of a polyhedral complex with strata (i) (; ), i = 1; 2; ::::d,
dim((i) (; )) = i, where (d) (; ) = f(; )g and, inductively, (i 1) (; )
consists of all faces of polyhedrons in (i) (; ). (By a face of a k-dimensional
convex polyhedron we understand a maximal (k 1)-dimensional convex polyhedron contained in its relative boundary.) Fix a point v 2 (; ). The support of v in (; ) is the unique polyhedron in the complex that contains v
and is of least dimension among such, denote it by F. Observe that v 2 ri(F ).
Let G(F ) be the graph obtained by the following pruning of G: an edge e
of G is left in G(F ) , if and only if there is a loop in G passing through e
that satises () 2 F . For the corresponding subshift of nite type we write
(F ) : (F ) ! (F ). The collection of all thus obtained (F ) 's (as we vary
v in (; )) together with the obvious inclusions between them forms what
is called in [MT91] a scaold of subshifts of nite type. Note that subshifts
(F ) need not to be transitive, but this adds only a trivial complication to the
considerations (recall the remark after Theorem 2.2.1).
A simple argument in [MT91] shows that any loop in G(F ) must have
() 2 F . One can easily deduce then that
((F ) ; ) = F:
It is also observed in [MT91] that any ergodic measure for which
is supported on (F ); consequently,
hverg (; ) = hverg ((F ) ; ); for v 2 F:
R d 2 F
(2:23)
63
As a corollary, we get the following extension of Theorem 2.2.2 to v 62
ri((; )).
Theorem 2.2.4 Let v 2 Rd be such that hverg (; ) 0 (i.e. v = R d, for
some ergodic probability measure ). For 0 < < 1, there exists a compact
-invariant set K such that htop (jK ) hverg (; ) and (jK ; ) = v.
In fact, K is such that, for some constant C , kSn(; )(x) nv k C , x 2 K ,
n 2 N.
R d for an ergodic measure , then v belongs to (; ). Let
F be the support of v in (; ). If F is a point, then one can take for K the
whole subshift (F ) because, by (2.23), hverg (; ) = hverg ((F ); ). In general
(v) (F )
v 2 ri(F ), so, by (v) of Th. 2.2.1, hverg (; ) = htop
( ; ), and we nd
K (F ) by applying Theorem 2.2.2 to the subshift (F ) .
Q.E.D.
Proof. If v =
(v)
The behavior of htop
upon passing from to the subshift (F ) is given by
the following proposition.
Proposition 2.2.2 For v 2 (; ) with support F (; ), we have
(v)
htop
(; ) = convex hull
(v) (F )
htop
( ; ) :
(2:24)
(If (F ) is transitive, then the convex hull is superuous.) We will give
two proofs of the proposition.
Short proof. Recall that is expansive making hvmeas (; ) upper semi-contin-
(v)
uous (cf. [Wal82]). Thus, since htop
(; ) is convex, Corollary 2.1.2 yields
(v)
htop
(; ) = hvmeas(; ) = convex hull
hverg (; ) :
64
Using (2.23), we can write
(v) (F )
(v)
hverg (; ) = hverg ((F ) ; ) htop
( ; ) htop
(; ):
Taking the convex hull of each function above nishes the argument. Q.E.D.
Note that ultimately the above proof owes its eciency to nontrivial results of ergodic theory. For a reader ready to fend o supercial technicalities,
we supply the following argument unveiling elementary mechanism standing
behind the proposition.
(v)
(v) (F )
Long elementary proof. Clearly htop
(; ) htop
( ; ); and so, by convexity
(v)
(v)
(v) (F )
of htop
(; ), we have htop
(; ) convex hull htop
( ; ) . We will
prove the opposite inequality by showing that, for arbitrary 0 < < 1,
(v)
htop
(; ) convex hull
(v) (F )
htop
( ; ) :
It is enough to consider the case when F is a face of (; ). Let V be
the ane space spanned by F . Set
(
d := min dist ((); V ) :
X
)
26 V; elementary loop in G :
(v)
When htop
(; ) = 0, there is nothing to prove. Otherwise, observe that
one can nd > 0 such that, for suciently large n 2 N,
(v)
nN (n; n)#Gn exp (2n) exp (1 )htop
(; )n ;
(2:25)
where N(k; m) is the binomial Newton symbol. Also, let W F be discrete
and such that the balls fB (w)gw2W cover F = ((F ); ).
65
Pick 0 < r0 < and k0 2 N so that:
if 0 is a path in G(F ) with l( 0 ) k0, then (1=l( 0)) P0 2 Sw2W Br (w),
and, for w 2 W ,
0
w) (F )
( ; ) + )k ; for k > k0:
sBr (w) ((F ); ; 1; k) exp (h(top
0
(2:26)
(See the page 30 for denition of sBr (w)((F ); ; 1; k).)
Now, take r > 0 small enough to have
Nr(k0 + 1)=d ;
(2:27)
where N is the number of vertices in G.
Consider for a while a xed loop in G of length l( ) = n and with
( ) = (1=l( )) P 2 Br (v). We will show that \most" of lies in G(F ).
As described in the argument for (i) of Theorem 2.2.1, one can rearrange
the edges of to get elementary loops 1; :::; p such that P = Ppi=1 Pi .
Note that
dist
X
! X
; nV =
i
l(i)dist((i ); V ) #fi : (i) 62 V g:
Since v 2 F V , P 2 Bnr (nv) implies that at most nr=d of i's have
(i) 62 V . Since the length of an elementary loop is by denition less than
N , at most Nnr=d of the edges in can belong to G n G(F ) . After deleting
these edges, the path splits into at most nNr=d paths in G(F ), of which those
longer than k0 are denoted by 1, ... q , q nNr=d, in the order of appearance
along . The combined length of i's is greater than n nNr=d nNrk0 =d =
n nNr(k0 + 1)=d (1 )n, where we used (2.27) for the last inequality.
66
One of the consequences is the following basic estimate
X 0
(1=l( ))
k (x)k:
:= 2 sup
x2
i=1 i
j =1
Furthermore, with each i associate wi 2 W chosen so that
(1=l(i)) X wi r0 ; i = 1; :::q:
i
q
q X
X
X
l
(
))
(1
=
j
This and (2.28) yield
(1=
q
X
j =1
l(j ))
q
X
i=1
l(i)wi 2 Br++0 (v):
(2:28)
(2:29)
(2:30)
We claim that relatively few loops of length n as considered above can
share the same i's (in the same order and positions along ) and also have a
coinciding vector of wi's associated with i's. Indeed, by basic combinatorics,
the number of possibilities for the choice of the deleted elements is bounded
by
n N(n; n)(#G)n
and the number of choices of wi's by
N(n; #W ) exp(n);
where the right most inequality holds for large n.
(v)
By denition of htop
(; ) (see (2.13)), there are arbitrarily large n 2 N
(v)
with at least exp((htop
(; ) )n) dierent 's as considered above. The two
combinatorial estimates and \the pigeon-hole principle" guarantee that some
selection of wi's and the deleted elements (and thus also of i's) is shared by
at least
(v)
(v)
exp (htop
(; ) )n =n N(n; n)(#G)n exp(n) exp htop
(; )n
67
dierent 's, where the inequality is due to the choice of , see (2.25).
For each i = 1; :::q, denote by Si a collection of all dierent i's showing in
(v)
those 's. The above estimate implies that #S1 :::#Sq exp htop
(; )n .
Also, by the choice of r0 and k0 and using (2.29), we get
(wi ) (F )
#Si exp (htop
( ; ) + )l(i) ; i = 1; :::q:
Consequently, after taking the logarithms and limsup with n ! 1, we get
(v)
(;
htop
1 q
0 q
X
X
(wi ) (F )
( ; ) + ) @1= l(j )A l(i)htop
j =1
i=1
and thus, if w := 1= Pqj=1 l(j ) Pqj=1 l(i)wi, then
(v)
htop
(; ) convex hull
w) ( (F ) ; )
h(top
:
By (2.30), when is shrunk to zero, w approaches v, and the desired inequality
follows by lower semi-continuity of the right side above in the argument w.
Q.E.D.
Chapter 3
Topological entropy at the rotation vector for
torus maps
The rotation set, in the case of torus homeomorphisms, is very far from
being a complete topological invariant; this makes sensible the search for ner
invariants. While the presence of a particular vector v in the rotation set
merely tells us that there is \some" dynamics with this asymptotic average
displacement, one may try to measure the abundance of those dynamics. Realizing this philosophy, we associate with v the corresponding \topological
entropy". We use quotation marks because, even though there is a well agreed
on denition of the topological entropy of a map ([Wal82]), there are many
ways of relativizing it to v. We will mainly use a simple topological concotion
(v)
htop
and its measure theoretic counterpart hverg , both of which are extensively
(v)
discussed in Chapter 2 and Appendix A. For the pseudo-Anosov maps, htop
and hverg coincide and are explicitly calculable once we know the Markov partition (Theorem 2.2.1). The pseudo-Anosov maps, our favorite examples, also
(v)
provide a way to give lower bounds for htop
for a general map.
68
69
In the rst section, we dene the relevant entropies and state the lower
bound they obey. In the second section, we translate the results of Section 2.2
to the case of the pseudo-Anosov maps. In the third section, combining the
main theorems of Chapter 1 and Chapter 2, we prove the lower bound. Last
(v)
section is devoted to a concrete example of a calculation of htop
.
3.1 Denitions and the main result
Recall that for f 2 H(T2 ) and its lift F 2 H~ (T2), F id descends to
the displacement function F : T2 ! R2. Note that (F ) = (f; F ). More
generally, given an invariant set K T2 , we will write (F jK ) for (f jK ; F ).
Denition 3.1.1 For v 2 R2, the topological entropy at the rotation vector v
is
(v) (F ) := h(v) (f; ) 2 f 1g [ [0; +1]:
htop
F
top
Similarly, we have
hverg (F ) := hverg (f; F ) 2 f 1g [ [0; +1]:
(v)
Our ultimate goal in this chapter is to bound entropies hverg (F ) and htop
(F )
from below in terms of the \size" of the rotation set and the relative distance
of v to its boundary.
To this end, we state the following denition. For a convex compact
K R2 with int(K ) 6= ;, we dene the tent function of K by
K (v) := inf fka vk=ka bk : v 2 convfa; bg; a; b 2 @K g ; v 2 int(K ):
70
Before we say more about the properties of K (v), we formulate our main
result. (For the denitions of A() and I (), used below, see Section 1.1.)
Theorem 3.1.1 (Main Lower Bound) There are universal constants C and
C 0 such that, if f 2 H(T2 ) is a dieomorphism, F 2 H~ (T2) is its lift and
v 2 int((F )), then
(v)
htop
(F ) hverg (F ) maxfC ln+ A((F )); C 0=I ((F ))g (F ) (v):
(3:1)
Moreover, there exists an invariant compact set T2 such that
(F j) = fvg
and
htop (f j) maxfC ln+ A((F )); C 0=I ((F ))g (F ) (v):
(3:2)
The proof of the theorem is given in Section 3.3; however, Section 3.2 is a
necessary prerequisite for it.
Let us now shed some light on K (v). First of all, it depends only on the
ane properties of K . There is however a trivial bound
K (v) dist(v; @K )=diam(K ):
The \inmum" in the denition of K (v) is achieved when a and b are
antipodal, meaning that there is a pair of parallel lines, one tangent to @K
at a, and the other tangent to @K at b. Indeed, by compactness of @K and
continuity of the norm, there are a and b such that v 2 convfa; bg and K (v) =
ka vk=ka bk. Figure 3.1 below gives an idea of what would go wrong if a
and b were not antipodal.
71
a
v
a2
a1
b2
b
b1
Figure 3.1: When a and b are not antipodal.
From Figure 3.1, note that ka1 vk < ka2 vk and kv b1 k > kv b2 k,
so ka1 vk=kv b1 k < ka2 vk=kv b2 k = ka vk=kv bk. Thus ka1
vk=ka1 b1k < ka vk=ka bk, a contradiction.
Finally, observe that K (v) is concave in v. In fact, we have
(
K (v) = inf g(v) : g : K !
R+
)
concave with sup g(x) = 1 :
x2K
(3:3)
(This is why we call K (v) the tent function of K .) For a proof, x v 2 K . If
g : K ! R+ is concave with sup g = 1, then one can take a; b 2 @K such that
convfa; bg contains v and a point w 2 K where g(w) = 1. By concavity,
g(v) kkvw wbkk g(w) + kkwv bbkk 0 = ka vk=ka bk K (v):
On the other hand, if K (v) = ka vk=ka bk, then there is a g such that
g(v) ka vk=ka bk; simply consider the two parallel tangent lines through
a and b and take g linear and constant on each of them.
72
3.2 The central example: pseudo-Anosov maps
Throughout this section let f 2 H(T2) be a map that is pseudo-Anosov
rel a nite set P and let F : R2 ! R2 be its lift to the universal cover. An
abundance of rotation sets is exhibited by such examples. Indeed, we know
that, given a convex polygon 0 with vertices in Q2, there is F 2 H~ (T2) such
that (F ) = 0 ([Kwa93]). By Theorem 1.2.1 and the subsequent remark, we
have G 2 H~ (T2) that is a lift of a pseudo-Anosov rel a nite set and has
(G) = 0 .
(v) (F ) and hv (F ) in the pseudo-Anosov case is
Eective treatment of htop
erg
made possible by the method of symbolic dynamics.
Proposition 3.2.1 ([FLP79]) There exists a Markov partition R = fR1 ; :::;
RN g for f . If G is a graph with the set of vertices f1; :::; N g that has an edge
from i to j whenever f (Ri ) \ int(Rj ) =
6 ;, then the subshift of nite type (; )
associated with G (see sec. 2.2) is mixing and factors onto f . More precisely,
for any x = (xi)i2Z 2 the intersection Ti2Z f i(Rxi ) consists of a single point,
denoted by p(x), and thus dened map p : ! T2 has the following properties:
(i) p = f p,
(ii) p is continuous and surjective,
(iii) p is nite-to-one,
(iv) p is 1-1 onto a topologically residual set consisting of all the points which
full orbits never hit the boundary of the Markov partition.
The above proposition gives a rather complete description of the dynamics
of f in terms of the shift . To keep track of the averages of the observable 73
in the symbolic model, we need to associate with every edge (i; j ) of the graph
G a vector weight i;j , reecting the displacement in the universal cover that
is inicted on lifts of points in Ri that are mapped into Rj . One way to do
that (taken from [Kwa92]) is as follows. Select a point zi in each Markov box
Ri, i = 1; :::N . If (i; j ) 2 G, then f (Ri) \ int(Rj ) 6= ;, so there is a pair of
corresponding lifts such that F (R~i) \ int(R~j ) 6= ;; dene i;j := z~i z~j ; where
z~i 2 R~i and z~j 2 R~j are the lifts of zi and zj respectively. This denition does
not depend on the choice of the lift R~i. Also, recall that the weights i;j can
be thought of as an observable : ! R2 given by ((xi)i2Z) := x ;x .
0
1
Claim 3.2.1 For x = (xi )i2Z 2 and any n 2 N, we have
kSn(f; F )(p(x)) Sn(; )(x)k 2 diam(R):
In fact, F p and
are cohomologous.
Proof. Having xed a lift R~x of Rx , let R~ x be the lift of Rx such that
z~1 z~0 = x ;x , let R~x be the lift of Rx such that z~2 z~1 = x ;x , etc. Thus
dened sequence R~xj (j = 0; 1; 2:::), by the denition of , has the following
0
0
1
2
0
1
1
2
1
2
property: if q~ 2 R~x is a lift of q = p(x), then F i(~q) 2 R~xj , j = 0; 1; 2:::. The
inequality follows because Sn(f; F )(q) = F n(~q) q~ and Sn(; )(x) = z~xn z~x ,
whereas z~j 2 R~xj , j 2 N. Moreover, if u(x) := q~ z~x , then F p = u u,
making F p and cohomologous.
Q.E.D.
0
0
0
Proposition 3.2.2 Suppose that F 2 H~ (T2 ) descends to f 2 H(T2 ) that is
pseudo-Anosov rel a nite set. If (; ) is the corresponding subshift of nite
74
type and
is the corresponding observable on , then
(F ) = (f; F ) = (; );
and, for any v 2 int((F )),
(v)
(v)
htop
(F ) = hverg (F ) = htop
(; ) = hverg (; ):
(3:4)
Moreover,
(i) there is a fully supported ergodic probability measure (v) such that h v (f ) =
R
hverg (F ) and F d(v) = v, v 2 int((F )),
(v)
(ii) htop
(F ) is strictly concave and real analytic in v on int((F )).
( )
Remark. In fact hverg (F ) = hverg (; ) for all v 2 (F ).
Proof Proposition 3.2.2. Part (ii) follows from the main assertion by (ii) of
Theorem 2.2.1. The statement about the rotation sets follows easily from
(i) and (ii) of Proposition 3.2.1 coupled with Claim 3.2.1. To deal with the
entropies, consider an invariant -ergodic probability measure . It pushes
R
R
forward to := p( ), which is ergodic with respect to f ; and d = F d
because p is cohomologous to F . Moreover, if is fully supported, then
we claim that
h (f ) = h ():
(3:5)
This is standard. Observe that is also fully supported; and, by ergodicity,
the full measure is carried on the invariant residual set on which p is 1-1.
Hence, the measure theoretical systems (T2; f; ) and (; ; ) are isomorphic
via p.
75
Now x v 2 int((F )). Taking for the Gibbs state averaging
to v as
in (v) of Theorem 2.2.1, we see that (3.5) yields
(v)
(v)
htop
(f; F ) hverg (f; F ) h(f ) = h () = hverg (; ) = htop
(; ):
(v)
(v)
(f; F ) htop
(; ) (see
On the other hand, since f is a factor of , htop
Proposition 2.1.4), so all the quantities above must be equal. This proves
(3.4), and we see that (v) := is as stipulated by (i).
Q.E.D.
Proof of the remark. Since f is a factor of we have hverg (f; F ) hverg (; )
(see Proposition 2.1.4). To prove the opposite inequality, take a -ergodic
R
probability measure with d = v and set := p ( ). We need to show
that h (f ) h (). Let E T2 be the full orbit of the union of the stable
boundaries and the unstable boundaries of the Markov boxes, denoted by @ Rs
and @ Ru respectively. If (E ) = 0, then h (f ) = h () (as in the proof of
the proposition). Otherwise, by ergodicity, (E ) = 1. Since @ Rs is forward
invariant under f and @ Ru is backward invariant under f , we have either
(@ Rs ) = 1 or (@ Ru ) = 1. Say, we have the rst case (for the other use f 1).
Then, h (f ) htop (f j@Rs ) = 0, where the vanishing of the topological entropy
follows from the fact that @ Rs is a collection of smooth arcs (cf. [Bow78b])
with the length strictly contracted by f ([FLP79]). Since p is nite-to-one
htop (jp (@Rs ) ) = 0 (see Th. 17 in [Bow71]); thus h () htop (jp (@Rs ) ) = 0.
Our claim follows.
Q.E.D.
1
1
Using the scaold construction (see the subsection 2.2.3 and Theorem 2.2.2),
one can verify the following: for v 62 int((F )), if there are any measures
76
(v)
realizing htop
(; ), then they are the Gibbs states for the smaller subshift of
nite type (F ) that corresponds to the support F of v in (F ). If is such
a Gibbs state realizing hverg (; ), then the argument above shows that p ( )
realizes hverg (f; F ).
As a corollary of Theorem 2.2.2 we get the following.
Theorem 3.2.1 Suppose that f 2 H(T2) is pseudo-Anosov rel a nite set, F
is its lift and v 2 int((F )). For any 0 < < 1, there is a compact set Q T2
(v) (F ).
invariant under f such that (F jQ) = fvg and htop (f jQ) > herg
Proof of Theorem 3.2.1. Take Q = p(K ), where K is as in Theorem
2.2.2. The fact that (F jQ) = (jK ; ) follows from Claim 3.2.1. The map p
is nite-to-one, so we have htop (jK ) = htop (f jQ) (by Th. 17 in [Bow71] and
the fact that entropy on a nite set must be zero).
Q.E.D.
Our focus on the case v 2 int((F )) is partially justied by the following
fact.
1
Proposition 3.2.3 If f 2 H(T2) is pseudo-Anosov rel a nite set and F is
its lift, then int((F )) =
6 ;.
Proof of Proposition 3.2.3. Passing to an iterate of F and perhaps postcom-
posing it with a deck transformation, we can assume that F has a xed point
x = F (x) 2 R2. Fix a Markov partition for f . Denote by R the lift of the
Markov box containing x. We will show that there are three non-collinear
1 This is essentially due to
Fried, compare Theorem H in [Fri82a].
77
vectors in (F ). Think of T22 := R2=(2Z)2, a 4-to-1 covering of T2 = R2=Z2 .
The map F factors to f2 : T22 ! T22, which is also pseudo-Anosov rel a
nite set. By transitivity of the unstable foliation ([FLP79]), the unstable
leaf through the projection of x to T22 intersects the projections of int(R),
int(R) + (1; 0), and int(R) + (0; 1). It follows that, for some large n 2 N,
we have F n(R) \ int(R) + k0 6= ;, F n(R) \ int(R) + k1 + (0; 1) 6= ; and
F n(R) \ int(R)+ k0 +(1; 0) 6= ;, where ki 2 (2Z)2 , i = 1; 2; 3. This puts k0=N ,
(k1 +(1; 0))=N and (k2 +(0; 1))=N in the rotation set of F . These three vectors
are not collinear because the vectors k1 k0 + (1; 0) and k2 k0 + (0; 1) have
the corresponding coordinates of opposite parity and so can not be linearly
dependent.
Q.E.D.
3.3 Proof of the main result
We have f 2 H(T2 ) with a lift F 2 H~ (T2). Assume that int((F )) 6= ;;
otherwise, there is nothing to prove. Fix v 2 int((F )) and take arbitrary
> 0. One can choose a convex polygon 0 int((F )) with vertices in Q2 so
that A(0) = A((F )), I (0 ) = I ((F )) and 0 (v) (F ) (v) . By Theorem
1.2.1, there is a nite set P (a union of primitive periodic orbits) and a map
g 2 H(T2 ) such that: f and g are isotopic rel P , g is pseudo-Anosov rel P , g
has a lift G 2 H~ (T2) with (G) 0 .
The following theorem asserts that the dynamics of g are fully reected
by f . It is a variation on Handel's global shadowing result in [Han85] that is
most closely related to Theorem 3.2 in [Boy10]. (Also, this is the only place
78
where we are forced to use smoothness of f . One should be able to remove
this obstacle.)
Theorem 3.3.1 If f 2 H(T2) is a dieomorphism, P is a nite invariant
set and g : T2 ! T2 is pseudo-Anosov rel P and isotopic to f rel P , then
there is a compact invariant set Y T2 and a surjection h : Y ! T2 that is
homotopic to the inclusion map Y ,! T2 and semi-conjugates f jY to g (i.e.
h f jY = g h).
The map g and the factor map h from the theorem can be lifted to
G 2 H~ (T2 ) and an equivariant H : 1(Y ) ! R2 respectively, so that
H F j (Y ) = G H . In particular, for x 2 Y , we have
1
kSn(f; F )(x) Sn(g; G h)(x)k D := 2 sup kH (y) yk; n 2 N: (3:6)
y2 1 (Y )
Indeed, if x~ is a lift of x, the left side is equal to
k(F n(~x) x~) (Gn H (~x) H (~x))k = kF n(~x) H F n(~x) + H (~x) x~k:
W need the following easy corollaries, which proof we postpone.
Corollary 3.3.1 Under the assumptions of Theorem 3.3.1, we have:
(i) for any compact invariant Q T2, the set W := h 1 (Q) is compact invariant, (F jW ) = (GjQ) and htop (f jW ) htop (gjQ);
(ii) for any 2 E (T2 ; g), there is 2 E (T2 ; f ), such that := h ( ),
R d = R d and h (f ) h (g);
G
F
(v)
(v)
(iii) for any v 2 Rd , htop
(F ) htop
(G) and hverg (F ) hverg (G);
(iv) (G) (F ).
79
Proof of Theorem 3.1.1. To prove (3.1) of the theorem, observe that, by (ii)
(v)
of Theorem 2.2.1, htop
(G) is a concave function of v; thus, from (3.3) and (iii)
of Proposition 2.1.1, we have
(v)
x) (G) : x 2 R2 g (v ) = h (g ) (v ):
htop
(G) supfh(top
top
(G)
(G)
Using Theorem 1.1.1 and (iii) of Corollary 3.3.1, we obtain
(v)
(v)
htop
(F ) htop
(G) maxfC ln+ A((G)); C 0=I ((G))g (G) (v);
and the analogous inequality for hverg . Because 0 (G) (F ) and (G) (v) (F ) (v) (where is arbitrary), the inequality (3.1) follows. The assertion
(3.2) is an immediate consequence of Theorem 3.2.1 via (i) of Corollary 3.3.1.
Q.E.D.
Proof of Corollary 3.3.1. Part (i): the equality of the rotation sets follows from
(3.6); the entropy assertion is standard ([Wal82]). Part (iii): use (3.6) to note
(v)
that, in calculating htop
(F ) and hverg (F ), one can use G h in the place of F
and then recall the quotient rules | Proposition 2.1.2 and Proposition 2.1.4.
The part (ii) can be extracted from the proof of Proposition 2.1.4. The part
(iv) is immediate from (3.6). (It also follows from (iii) by (ii) of Proposition
2.1.1.)
Q.E.D.
Proof of Theorem 3.3.1. We will show how to reduce the result to Theorem
3.2 (and its proof) in [Boy10].
We start with a construction taken from [Bow78a] and rst used in a similar context in [LM91]. Let M be the compact surface with boundary obtained
80
by blowing up the points of P to small circular discs and then by removing its
interiors. Denote by c : M ! T2 a continuous map that collapses the boundary circles of M to the corresponding points of P and is 1-1 elsewhere. Using
the derivative of f at points of P , one constructs f1 : M ! M that factors
to f via c, i.e. c f1 = f c. Now, let g1 : M ! M be Boyland's condensed
homeomorphism in the isotopy class of f1 (see Case 5, p. 13 in [Boy10]). In
our case, g1 is just a pseudo-Anosov map with some standardized behavior
at the boundary | it is called (in [Boy10]) boundary-adjusted pA map. In
particular, g1 factors to a map that is pseudo-Anosov rel P and (after perhaps
some conjugation) coincides with g, i.e. c g1 = g c. Theorem 3.2 in [Boy10]
yields a compact invariant set Y1 M and a surjective map h1 : Y ! M
(homotopic to the inclusion) such that h1 f1 = g1 h1. The map h1 needs
not to be continuous, however c h1 is.
Put Y = c(Y1). Because the map f1 was obtained as a blow up of f , it
is a tautology to say that it is uniformly continuous on int(M ) in the metric
induced from T2 by c : int(M ) ! T2. Now, one has to trace word by word
Boyland's proof of uniform continuity of c h1 on int(M ) while substituting
in the place of the standard metric on int(M ) the one induced by c. As a
result, we get uniform continuity of c h1 in this new metric. It follows that
h := c h1 c 1 : Y ! T2 is a well dened and continuous map. This map
has all the required properties.
Q.E.D.
81
3.4 Example
(v)
We calculate htop
numerically for a concrete example of a pseudo-Anosov
map. This map is derived from a map in H(T2 ) via the method from Section
1.2 coupled with the Bestvina-Handel algorithm ([BH92]). Our discussion also
(v)
illustrates how results of Section 3.2 can directly bare on htop
when dealing
with a single explicitly dened map.
Let H; V 2 H~ (T2) be given by
H (x; y) := (x + j sin(y)j; y);
V (x; y) := (x; y + j sin(x)j);
and let h and v be the corresponding maps in H(T2 ). Consider the composition
f := v h 2 H~ (T2) and its lift F := V H 2 H~ (T2 ). Note that F (0; 0) = (0; 0),
F (1=2; 0) = (1=2; 1), F (0; 1=2) = (1; 1=2), and F (1=2; 1=2) = (3=2; 3=2). Thus
the square Q := f(x; y) : 0 x; y 1g is contained in (F ). In fact, we have
(F ) = Q because, by writing F = G h + H , one can verify that the range
of the displacement function F sits in Q.
Now, let P T2 consist of the three xed points of f that have (0; 0),
(1=2; 0), (0; 1=2) for their lifts. The isotopy class of f rel P is represented by
a map g that is pseudo-Anosov rel P (cf. Theorem 1.2.1). To nd out about
g, we use the algorithm by Bestvina and Handel.
The surface T2 n P deformation retracts to its spine consisting of the union
of the oriented loops ; ; ; , marked on Figure 3.2. To record the isotopy
classes of h; v; f , we postcompose them with the retraction and look at the
82
resulting transformation of the spine into itself. On Figure 3.2, the dotted
lines indicate the images of and under the resulting action on the spine for
h. Alternatively, we write h# : 7! , 7! , 7! , 7! , where the
bar over an oriented path switches the orientation.
y
γ
δ
γ
α
β
α
β
γ
δ
δ
α
α
β
β
γ
δ
x
Figure 3.2: The map h# on the spine.
Similarly, for v, we get v# : 7! , 7! , 7! , 7! . For
the composition f = v h, we get f# : 7! , 7! , 7! ,
7! .
It turns out that action of f# on the spine is not ecient; this means
that, as we apply f# repeatedly to arcs making up the spine, we get paths
that backtrack (see [BH92] for the denition). Bestvina and Handel describe
moves that transform the map f# and the spine leading to an ecient map
in a nite number of steps. The implementation of the algorithm, in our case,
involves nine steps and is more tedious then instructive, so we skip it. The
result is a graph contained in T2 n P (see Figure 3.3), which is a deformation
retract of T2 n P , and a transformation g1 of this graph that is ecient.
83
c’
y
h’
f’
c
b’
g’
e’
f
e
a
d
h
g
b
x
Figure 3.3: The ecient map g1.
The marked arcs are mapped by g1 as follows: a 7! a0 = a, b 7! b0 = b,
c 7! c0 = ghbh, d 7! d0 = cfe, e 7! e0 = ef , f 7! f 0 = cfg, g 7! g0 = dae,
h 7! h0 = fgh. As described in [BH92], the graph can be smoothed into a train
track that carries a pseudo-Anosov map g which deforms to g1. The Markov
boxes for g can be obtained by thickening the arcs c; d; e; f; g; h to rectangles.
Actually, Figure 3.3 depicts not only g but a certain lift G of g. To determine the rotation set (G), we read o the gure the displacements under G:
a 7! a
b 7! b + ( 1; 1)
c 7! g + ( 1; 1) h + ( 1; 1) b + ( 1; 1) h + ( 1; 1);
d 7! c f e;
84
e 7! e f;
f 7! c f + ( 1; 0) g + ( 1; 1);
g 7! d + ( 1; 0) a + ( 1; 0) e + ( 1; 0);
h 7! f + ( 1; 0) g + ( 1; 1) h + ( 1; 1):
Tying this with the material in Section 2.2, we draw the weighted transition graph of the corresponding subshift of nite type | see Figure 3.4. (For
clarity we skip weights (0; 0).)
(-1,1)
(-1,1)
b
a
h
(-1,0)
(-1,1)
c
g
(-1,1)
(-1,0)
(-1,0)
(-1,1)
(-1,0)
f
d
e
Figure 3.4: The transition graph.
It is apparent from this graph that (by Proposition 3.2.2 and (i) of Proposition 2.2.1)
(G) = convf(0; 0); ( 1; 0); ( 1; 1)g:
Note that the map F (that we started with) is equivariantly isotopic rel 1 (P )
to G + (1; 0), not G. By Corollary 3.3.1, we have
(v)
(v+(1;0))
htop
(F ) htop
(G); v 2 R2 :
85
(v)
To calculate htop
(G), we extract from the graph its transfer matrix, see
below. (The columns and rows, counted starting at the upper left corner,
correspond to c, d, e, f , g, h. We skip a and b because they do not correspond
to any Markov boxes.)
0
BB
BB 0
BB
BB
BB 1
BB
BB
BB
BB 0
A := B
BB
BB
BB 1
BB
BB
BB 0
BB
BB
B@
0
0 0 0 xy 2xy
0 1 1 0
0
0 1 1 0
0
0 0 x xy
0
x x 0 0
0
0 0 x xy xy
1
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC ;
CC
CC
CC
CC
CC
CC
CC
CC
A
(3:7)
where x = exp( s) and y := exp t, (s; t) 2 R2. The characteristic polynomial
of A is p(x; y; z) := det(A zI ) and can be found to be equal to
p(x; y; z) = x4 y2 x3 yz x3y2z x4y2z + x2 yz2 + x3 yz2 + x3y2z2
6x2yz3 + xz4 + xyz4 + x2 yz4 z5 xz5 xyz5 + z6 :
(u;v)
Given a rotation vector (u; v) 2 int((G)), a way to nd htop
(G) is
through solving the following system of equations
p(x; y; z) = 0;
@p (x; y; z) = x @p (x; y; z);
vz @z
@x
(3.8)
(3.9)
86
@p (x; y; z) =
uz @z
@p (x; y; z):
y @y
(3.10)
Provided the right solution (x; y; z) is taken, we have
(v;u)
htop
(G) = ln(z xv y u):
(3:11)
s;t) = ln z , where z is the leading
Indeed, by (iii) of Theorem 2.2.1, p(top
positive root of the rst equation. By (iv) of Theorem 2.2.1,
(v;u) (G) = supfsv + tu ln z : s; t 2 Rg;
htop
(3:12)
where the supremum is achieved when v = @ (ln z)=@s = x(@z=@x)=z and
u = @ (ln z)=@t = y(@z=@y)=z. This was used to obtain the two last equations
of the system by taking partial derivatives of the rst one with respect to s,
for one, and with respect to t, for the other.
The system (3.8) may have many solutions. Continuation of a known
solution is a way to select the right one. For example, one can start with x = 1
and y = 1 nding the leading positive root of p(1; 1; z) to be approximately
2:6103. Thus
htop (g) = max
h(v;u) (G) = ln 2:6103:
(v;u) top
The corresponding rotation vector, calculated from the measure of maximal
entropy (see (v) of Theorem 2.2.1), is
(vmax ; umax) = ( 2=3; 1=3):
(v;u)
Changing (u; v) away from (vmax ; umax), we may follow htop
(G). In this way,
using Maple, we obtained for example h(top0:42;0:33333) 1:74690.
Chapter 4
Rotation sets for the resonantly kicked linear
oscillator
As we have seen in the previous chapters, one can infer a lot about dynamics of a torus map whose rotation set has nonempty interior. We also
know that there is an abundance of such maps. (See the example in the section 1.1, Proposition 3.2.3, also [Kwa92].) However, our examples so far were
constructed ad hoc to get the rotation set with specic properties. In this
chapter we consider a model arising from plasma physics.
4.1 The physical context
Consider a particle of mass M and charge e moving in R3 lled with a
uniform vertical magnetic eld of induction B0. The force exerted by the eld
on the particle is proportional to the vector product of its velocity and the
eld. Thus the trajectories are curled into spirals with circular projections on
the horizontal plane. More precisely, if x; y; z 2 R are the coordinates of the
87
88
particle (z being the vertical one), then the Newtonian equations of motion
are
M x = eBc 0 y;_
M y = eBc 0 x;_
M z = 0;
where c is the speed of light. The vector of horizontal velocity (x;_ y_ ) rotates
with the Larmor frequency ! = eB0 =cM . The three momenta: pz := M z_ ,
py := M y_ + M!x, and px := M x_ M!y, are conserved. This simple behavior
can change radically in the presence of an external electric eld in the system.
We will restrict our attention to the case when the electric eld is aligned with
the x-axis and does not depend on y; z coordinates (cf. [ZZRSC86b]). These
assumptions reduce the number of essential degrees of freedom to one since
the momenta pz , py continue to be integrals of the motion. Furthermore, we
will require that the eld is a \tight" wave packet with amplitude E (x; t) =
P1 TE (x)(t Tn), where T > 0 is xed, E (x) is 2=k-periodic in x,
n= 1
and () is the Dirac delta function. The meaning of the formula is that the
packet interacts with the particle by boosting its x-velocity by Me TE (x) every
T seconds. Thus, if (x_ n; y_n) is the horizontal velocity prior to the kick at the
time nT (n 2 Z), then
(x_ n+1; y_n+1) = R x_ n + eT E py M y_n ; y_n
(4:1)
M
M!
where R is the rotation by the angle = !T . In this way, the analysis of the
physical model reduces to investigation of dynamics on R2 generated by the
89
mapping
F : (x_ n; y_n) 7! (x_ n+1 ; y_n+1):
This reduction was rst carried out in [ZZRSC86b].
What makes the map F interesting is that (even for small amplitudes of
E ) numerical experiments and physical heuristic suggest that it exhibits trajectories escaping towards the innity with a nonzero \average acceleration".
A signicant physical consequence is an unbounded diusion of plasma particles called also stochastical heating ([ZZRSC86b]): the particles increase their
kinetic energy (heat up) by exploiting the energy of the electric eld.
In general, there are no rigorous arguments for the existence of the above
behavior. A lot of attention is thus devoted to resonant cases when is rational
([ZZRSC86b, CSUZ87, ZZRSC86a]). Among those, the cases of choice are
= =2, = =3 or = 2=3 | the only possibilities if we require that
some positive iterate of F is a doubly-periodic map of R2, i.e. it commutes
with some faithful Z2 -action on R2. Moding out by the Z2 -action yields a torus
dieomorphism isotopic to the identity. In terms of the torus map, the presence
of stochastical heating corresponds to a nontrivial rotation set. Indeed, in
view of (4.1), the rotation set can be interpreted as the totality of asymptotic
average accelerations exhibited by the particles (with xed momentum py ).
In what follows we will analyze closer the case of = 2=3. (The cases
= =3; =2 can be treated analogously.) Our modest goal is to see that, for
rather high values of nonlinearity, the rotation set is indeed nontrivial. We
calculate it for special cases and provide crude estimate from below in the
90
general case. In the case when is \saw-tooth-like", we prove monotonicity
of the rotation set as a function of the amplitude of E (also for rather large
amplitudes). The more delicate part of our discussion is greatly simplied by
formulation of the problem in terms of a second order recurrence equation.
This simple reduction has not been taken up in this context before. We close
our discussion by mentioning a toy model in statistical mechanics governed by
the recurrence equation.
4.2 The mapping and its rotation set
The rotation R2=3 is linearly conjugated to the map
L : (u; v) 7! (u; u v); u; v 2 R:
By choosing the conjugacy that sends x_ -axis to u-axis and by rescaling the
variables, one conjugates F (as prescribed by (4.1)) to G given by
G : (u; v) 7! L(u + (v); v) = (v; u v (v));
(4:2)
py ) is 2 -periodic. The third iterate of G
where (v) = eT=M!k E ( kv + M!
commutes with the translations of R2 by vectors in (2Z)2, and so it factors
to a torus map
g 3 : T2 ! T2 :
Even though G and g3 depend on the choice of , we suppress this dependence
in the notation since it will be always clear which we are talking about.
From now on, if we do not state otherwise, is an arbitrary xed continuous
91
function of period 2. However, we will remember that naturally comes
embedded in the two parameter family
K; (x) := K(x + ); x 2 R;
(4:3)
where 2 R and K > 0 correspond to the momentum py and the strength of
the forcing respectively.
Fact 4.2.1 For any 2 R, replacing () with a shifted function ( + ) + 3
is equivalent to conjugating G by a translation (u; v) 7! (u ; v ). In
particular, the rotation set (G3 ) of G3 is unaltered by the shift.
For r 2 R, we will denote by Hex(r) the hexagon with vertices (r; 0),
L(r; 0) = (0; r), L2 (r; 0) = ( r; r). The following are the most basic
properties of the rotation set (G3).
Proposition 4.2.1 If G is dened by (4.2), then
(i) (G3 ) has a three-fold symmetry, namely (G3 ) = L((G3 )) = L2 ((G3 ));
(ii) if is odd, then (G3 ) = (G3 );
(iii) (G3 ) is contained in Hex(sup inf ).
Proof of (i). Let Q : R2 ! R2 be given by
(u; v) 7! (u + (v); v);
so that G = L Q. The displacement of a point p 2 R2 under n + 1 iterates
of G is
Gn+1(p) p = L Q Gn(p) L Gn(p) + L(Gn(p) p) + Lp p:
92
The rst and the last dierences on the right side are bounded, so (1=(n + 1))
(Gn+1(p) p) is asymptotically equal to L ((1=n) (Gn(p) p)), as n ! 1;
and thus (G3 ) = L((G3 )).
Q.E.D.
Proof of (ii). Just note that if is odd, then Q commutes with central sym-
metry p 7! p, and so does the whole map G.
Q.E.D.
Proof of (iii). Write the displacement of p 2 R2 under G3 as follows
G3(p) p = L Q G2(p) G2(p) + L2 Q G(p) G(p) + L3 Q(p) p :
The arguments of L; L2; L3 = id, above, belong to a segment I with endpoints
(inf ; 0) and (sup ; 0), so G3(p) p sits inside the algebraic (Minkowski) sum
L(I ) + L2 (I ) + I , which equals Hex(sup inf ).
Q.E.D.
Proposition 4.2.1, (ii) can be sharp, as it is shown by the example below.
For convenience, we note the following rst.
Fact 4.2.2 If (u; v) 2 R2 satises ( v u (v)) = (v) and := (v)
(u) 2 2Z, then (u; v) is a lift of a xed point of g3 and (G3 ; (u; v)) = (0; ).
Proof. We apply G to (u; v) three times
(u; v) 7! (v; u v (v))
7! ( u v (v); u + (v) ( v u (v)))
= ( u v (v); u) 7! (u; v + (v) (u)):
Q.E.D.
93
Actually, from the proof, one can see that the inverse of the fact is also true.
Example. Consider any normalized so that sup = K and inf = K (see
Fact 4.2.1). By Fact 4.2.2, if there are u1; u2; v1 ; v2 such that (v1) = (v2) =
( v1 u1) = K and (u1) = (u2) = ( v2 u2) = K , we conclude that
both (0; 2K ) are in the rotation set provided also K = 0 (mod ). By (i) of
Proposition 4.2.1, the rotation set is then actually equal to Hex(2K ). It is not
dicult to draw a graph of with the required placement of the maxima and
the minima. (Use for example u1 = =2, u2 = =6, v1 = =6, v2 = =2.)
(One can see that any suitable has to attain one of its global extrema at
least three times per period.)
Another consequence of Fact 4.2.2 is the following very crude lower bound
on the rotation set.
Proposition 4.2.2 If l := sup inf 24, then the triangle with vertices
Li (0; l=2), i = 1; 2; 3, is contained in the rotation set (G3 ).
Proof. Using Fact 4.2.1, we can shift the graph of so that there are v0 ; u0
with (v0) = sup 12 and (u0) = inf 12. Now, as we push v continuously away from v0 (in either direction), before (v) drops below sup =2,
the expression 2v (v) changes by at least 4 + 6 = 2, thus sweeping
the whole period. In particular, ( 2v (v)) sweeps, in the process, an
interval containing [inf ; inf =2]; and so, by the intermediate value theorem,
there is v such that l=2 = sup =2 inf =2 (v) ( 2v (v)) 2 2Z.
Set u = 2v (v), and see that (u; v) satises the assumption of Fact 4.2.2,
94
thus placing (0; (v) (u)) in the rotation set. By (i) of Proposition 4.2.1,
Li (0; (v) (u)), i = 1; 2, are in the rotation set as well.
Q.E.D.
4.3 The recurrence, higher period orbits, monotonicity
A convenient way of viewing a trajectory of a point (u; v) 2 R2 under G
is to label it as a sequence (xk )k2Z so that x0 = u, x1 = v and (xk ; xk+1) =
G(xk 1; xk ), for k 2 Z. It is then characterized by the following recurrence
equation
xk 1 + xk + (xk ) + xk+1 = 0; k 2 Z:
(4:4)
For n 2 3N, a periodic orbit of G of period n thus corresponds to x =
(xk )kn=01 2 Rn satisfying (4.4) with a cyclic indexing modulo n. For = 0, this
is when every point is xed by G3 , the two-dimensional space of solutions to
(4.4) is conveniently spanned by
e = (1; 1; 0; 1; 1; 0; :::; 1; 1; 0);
Se = (0; 1; 1; 0; 1; 1; :::; 0; 1; 1);
S 2e = ( 1; 0; 1; 1; 0; 1; :::; 1; 0; 1);
where S : Rn ! Rn shifts the coordinates cyclically to the right, and S 3e = e.
The orbits of G corresponding to periodic orbits of g3 on the torus with
nonzero rotation vector come from (xk )k2Z satisfying (4.4) together with the
periodicity condition xk+n xk = vk (mod 3) , where v1 ; v2; v3 2 2Z, n 2 3N. To
95
search for them, we may use the following scheme. (Below hx; yi := Pni=1 xiyi,
for x; y 2 Rn.)
Lemma 4.3.1 For n 2 3N, if z = (zk )kn=01 2 Rn satises
zk 1 + zk + (zk ) + zk+1 = k ; 0 k n 1;
(4:5)
where 2 (2 Z)n and the indexing is cyclic, then there is 2 (2 Z)Z satisfying
k 1 + k + k+1 = k (mod n); k 2 Z;
(4:6)
and the sequence xk := zk (mod n) + k is a solution to (4.4). The displacement
over a period is given by
xk+n xn = h; S k ei; k 2 Z;
(4:7)
so the corresponding rotation vector for G3 is (1=n) h; ei; h; Sei . Moreover,
one can obtain in this way all of (xk ) 2 RZ corresponding to orbits of G that
cover periodic orbits of g3 on the torus.
Proof. Solving (4.6) is straightforward: x 0 ; 1 2 2 Z and gure out the rest
of i's successively. That xk = zk (mod n) + k satisfy (4.4) follows from putting
together (4.5) and (4.6). To calculate the displacement, replace in (4.4) k with
k + 1 and subtract unaltered (4.4) to get
xk+3 xk = (xk+1) (xk+2) = (zk+1 (mod n)) (zk+2 (mod n)):
Sum over the range k; k + 3; :::; k + n to obtain
E
D
xk+n xk = ((zj (mod n)))jk=+kn+1; S k e = h; S k ei;
(4:8)
96
D
E
where we used the fact that (zj 1 (mod n) +zj (mod n)+zj+1 (mod n))jk=+kn+1; S k e =
0. Now suppose that (xk )k2Z covers a periodic orbit of period n 2 3N. Then
xk+n xk =: vk (mod 3) 2 2Z, and, because both xk 's and the shifted sequence xk+n's satisfy (4.4), the sequence (vk (mod 3) )k2Z obeys the linear equation vk 1 (mod 3) + vk (mod 3) + vk+1 (mod 3) = 0. This makes k (mod n) :=
xk 1 (mod n) xk (mod n) (xk (mod n) ) xk+1 (mod n), k 2 Z, well dened.
To satisfy (4.5), set zk := xk , k = 0; :::; n 1.
Q.E.D.
Solving (4.5) for large n is a daunting task regardless of our choice of .
We thus have little hope that the rotation set can be calculated exactly for a
continuum of values of the parameters K and in (4.3). A way to somehow
bridge the particular cases for which the rotation set is nontrivial leads through
monotonicity results as the one below.
We will call unimodal, if it has only two intervals of monotonicity over
its minimal period. We will refer to the endpoints of the maximal intervals of
monotonicity as critical points of .
Proposition 4.3.1 For a piecewise-smooth unimodal and odd with D :=
inf x j0(x)j 6, the rotation set of G3 corresponding to K is nondecreasing
in K for K 2 [1; +1). Actually, all periodic orbits of the torus map can be
continued as K increases.
The proof is based on the following \closing lemma".
Lemma 4.3.2 Fix > 0 and suppose that z = (zk )kn=01 2 Rn satises
zk 1 + zk + K(zk ) + zk+1 = k + k ;
97
where k 2 ( ; ). If, for some open intervals Ik that are free of the critical
points of and contain zk , k = 0; :::; n 1, we have
d := inf fj1 + K0 (t)j : t 2 Ik ; 0 k n 1g > 2;
(4:9)
B d d (zk + K(zk )) fz + K(z) : z 2 Ik g;
(4:10)
and
2
then there is z = (zk ) 2 Rn such that zk 2 Ik , and
zk 1 + zk + K(zk ) + zk+1 = k ; 0 k n 1:
(4:11)
Proof. We obtain z as the limit of a sequence of corrections of z , each di-
minishing the supremum of k 's by a denite multiplicative factor < 1. The
hypothesis (4:9) guarantees that zk + K(zk ) changes faster (in zk ) than the
\coupling term" zk 1 + zk+1, thus k can be eectively decreased by manipulating zk only. The hypothesis (4.10) secures enough \maneuvering space" to
perform all innitely many corrections. With this in mind, it will be convenient to think of d d 2 as follows: x 2 (0; 1) and let = 1 + 2=d < 1,
then
d = = 1 = + + 2 + ::::
d 2 2=d
1 (4:12)
We describe the rst correction. From (4.12), d d 2 , so, by (4.10),
B(zk + K(zk )) fz + K(z) : z 2 Ik g:
It follows that we can nd zk0 2 Ik , k = 0; :::; n 1, so that
jzk 1 + zk0 + K(zk0 ) + zk+1 + k j = (1 )jk j
(4:13)
98
and
jzk0 + K(zk0 ) zk K(zk )j = jk j :
(4:14)
The hypothesis (4.9) implies then that
jzk0 zk j d jk j:
Putting the two inequalities together we get
zk0 1 + zk0 + K(zk0 ) + zk0 +1 = k + 0k ;
for some 0k such that
j0 j (1
k
)jk j + d jk 1j + d jk+1j 1 + 2 d = :
In this way
n 1 0
0 = max
j j :
k=0 k
(4:15)
To repeat the procedure leading from z = (zk ) 2 Rn to z0 = (zk0 ) 2 Rn with
z0 as our new z, we need to verify (4.13) with zk and replaced by zk0 and 0
respectively. In view of (4.10) and (4.14), it suces to check that
+ 0 d d 2 :
This is however a consequence of (4.15) and (4.12). It should be clear now how
the series in (4.12) unfolds as we need to guarantee the possibility of carrying
out the innitely many consecutive steps.
Q.E.D.
Proof of Proposition 4.3.1. Fix K > 1. We want to prove that the rotation
set of G3 corresponding to K contains that corresponding to . Note that,
99
by (i) of Proposition 4.2.1, the rotation set is either equal to f(0; 0)g or it has
nonempty interior. In the latter case, by Theorem 0.0.2, the rotation set is the
closure of the totality of the rotation vectors of periodic orbits. Consequently,
it is enough to show the following: for each periodic orbit of the torus map
generated by , there is one with the same rotation vector for K. In view of
Lemma 4.3.1, it suces to nd, for z 2 Rn satisfying (4.5), a point z 2 Rn
satisfying (4.11).
Let M := sup = inf . Clearly, and K have the same intervals of
monotonicity, K has larger image, and
jK(x) (x)j (K 1)M; for all x 2 R:
(4:16)
Thus, if z 2 Rn satises (4.5) for some 2 (2Z)n, we can nd for each zk a
point yk in the same interval of monotonicity such that K(yk ) = (zk ). By
(4.16) and the assumption on the derivative
jzk yk j (K 1)M=(KD) (K 1)M=(6K ):
In this way
yk 1 + yk + K(yk ) + yk+1 = k + k ;
with jk j := 3(K 1)M=(6K ) = (K 1)M=(2K ).
We will use Lemma 4.3.2 to nd z 2 Rn that obeys (4.11). The hypothesis (4.9) is satised because d D 1 5. Let Ik be the monotonicity interval
containing yk , that is Ik = [ck ; c0k ] where ck and c0k are the critical point of closest to yk . To verify hypothesis (4.10), observe that image under K of any
100
maximal interval of monotonicity extends exactly (K 1)M beyond the endpoints of the corresponding image for . This and the fact that jK0j KD
yield
jK(ck ) + ck K(yk ) yk j jK(ck ) K(yk )j jck yk j 1
1
jK(ck ) K(yk )j 1 KD (K 1)M 1 KD ;
and the analogous inequality for c0k . Since
d
d 2
D 1 (K 1)M
D 3 2K ,
to satisfy
(4.10), we need DD 31 (K 2K1)M (K 1)M (1 1=(KD)). Dividing both sides
by (K 1)M and setting D = 6, we get a stronger inequality 325K 1 61K ,
which is true because K 1.
Q.E.D.
4.4 A mechanical toy-model
Finally, let us point out a mechanical toy-model governed by (4.4). For
(zk ) 2 RZ, interpret each zk , k 2 Z, as the position of a unit mass particle
along the real axis. Let the particles interact by putting a Hookean string
between every two with consecutive indices. Also, subject the particles to a
potential force with the potential U (x) = 32 x2 , where is the antiderivative of . In the resulting mechanical model, the net force Fk exerted
on the kth particle is the sum of the potential force U 0 (zk ) = 3zk (zk )
and the interaction forces (zk+1 zk ) and (zk 1 zk ) from the two neighbors.
Thus we have
Fk = zk 1 + zk + zk+1 + (zk );
101
and the solutions to (4.4) can be interpreted as the stationary congurations of
this mechanical system. The construction is somewhat parallel to one relating
twist maps on the annulus to the Aubry theory (see e.g. [Ban88, Ang90]).
However, for now, we have no interesting theorems that stem from this approach.
Appendix A
Elementary large deviations for topological
dynamics
We give a proof of Theorem 2.1.1 that is elementary, it makes no use of
ergodic theory. Our original inspiration for this approach stems from Theorem
II.6.1 in [Ell85]. However, we present here a point of view not taken up in
[Ell85] that yields a very simple argument for Theorem 2.1.1. It can also be
used to considerably simplify the proof of Theorem II.6.1 in [Ell85].
Recall that we have f : X ! X , where X is compact and f is continuous.
Also, the observable : X ! Rd is a continuous function. Since f and will
be xed thruought the rest of the appendix, we will suppress the dependence
on them in the most of our notation. In particular, we will write n(x) for
Sn(f; )(x), that is n(x) = (x) + ::: + (f n 1(x)). For s 2 Rd and n 2 N,
we have the Gibbs weight ascribed to every point x 2 X , namely sn(x) :=
exphn(x); si. Given a nite set S X , the Gibbs weight sn() determines a
measure sn supported on S and given by sn := Px2S sn(x) x, where x is a
unit mass at x.
102
103
We are interested in the limiting distribution of the averages of among
(; n)-separated orbits counted with respect to the Gibbs weights. (Theorem
2.1.1 pertains to the case s = 0.) The corresponding pressure function ps :
X ! R (cf. free energy in [Ell85]) is given by
(X
)
s
s
p (t) := lim
lim sup(1=n) ln sup
exphn(x); tin(x) : S (n; )-separated :
!0 n!1
x2S
Observe that
(t+s)
ps(t) = p0(t + s) = ptop
(f; ); s; t 2 Rd;
(A:1)
and so, for the convex conjugate p^s of ps, we have
p^s(z) = p^0 (z) hz; si; s; z 2 Rd:
(A:2)
We should also note that replacing \lim sup" with \lim inf" in the denition
of ps yields the same quantity (see (viii) of Theorem 9.4 in [Wal82]).
For a set E Rd, we generalize hEtop (f; ) and introduce
H+s (E ) := lim
lim sup(1=n) ln supfsn(S ) : S (; n)-separated
!0 n!1
with n(x)=n 2 E , for all x 2 S g;
H s (E ) := lim
lim inf(1=n) ln supfsn(S ) : S (; n)-separated
!0 n!1
with n(x)=n 2 E , for all x 2 S g:
Both H+s (E ) and H s (E ) take values in f 1g [ [0; +1]. Clearly, if E F Rd, then H+s = (E ) H+s = (F ). (Here H+s = stands for either of H+s and
H s .) One can also easily verify that, for E; F Rd,
H+s (E [ F ) = maxfH+s (E ); H+s (F )g;
(A:3)
104
and if H+s (E ) < H s (E [ F ), then H s (F ) = H s (E [ F ), what implies
H s (E ) H s (E [ F ) maxfH+s (E ); H s (F )g:
(A:4)
(v)
Mimicking the way htop
(f; ) arises from hEtop (f; ), we associate with H+s
and H s the following entropies at z 2 Rd:
hs+(z) := rlim
H s (B (z));
!0 + r
hs (z) := rlim
H s (Br (z)):
!0
(z)
Note that h0+(z) = htop
(f; ), z 2 Rd. In general, hs+ 6= hs (see the example
ahead and Proposition A.0.2). The function z 7! hs+= (z) is quite obviously
upper semi-continuous. Also, note that if D := convf(x) : x 2 X g, then
(1=n)n(x) 2 D, x 2 X , n 2 N. Consequently, for E Rd, if E \ D = ;,
then
H+s = (E ) = 1:
(A:5)
In particular, hs+= (z) = 1; for z 62 D. (Actually, one can take here D :=
(f; ) | see Proposition 2.1.1.)
The dependence of hs+= (z) on s is analogous to that of p^s, namely
hs+= (z) = hz; si + h0+= (z); z; s 2 Rd :
(A:6)
This is because, if r > 0 and S X is a (; n)-separated set with (1=n)n(x) 2
Br (z) for x 2 S , then
exp ( nr) jsn(x)= exphnz; sij exp (nr); s 2 Rd;
and consequently,
exp ( 2nr) j exphnz; si0n(S )=sn(S )j exp (2nr):
105
Lemma A.0.1 For any s 2 Rd,
(a) if U Rd is open, then
H+s = (U ) supfhs+= (z) : z 2 U g;
(b) if V Rd is closed, then
H+s (V ) maxfhs+(z) : z 2 V g:
Proof. Part (a) is trivial. We prove (b) now. Let D be the convex hull
of the image of . From (A.5), we see that H+s (V ) = H+s (V \ D). Thus
we can assume that V is compact. Now, set V0 := V , and cover V0 with a
nite family A of its compact subsets of diameter less than 1. By (A.3), we
have H+s (V0) = maxfH+s (A) : A 2 Ag; denote by V1 an element of A such
that H+s (V0) = H+s (V1). Analogously, covering V1 with its compact subsets of
diameter less than 1=2, we get V2 V1 with H+s (V1) = H+s (V2). By continuing
this procedure, we get a nested sequence of compact sets V0 V1 V2 V3 :::
with diameters diam(Vn) < 1=n and H+s (Vn) = H+s (V ), n 2 N. Thus Tn2N Vn
consists of a point, and one can easily verify that, if v is this point, then
H+s (v) H+s (V ).
Q.E.D.
The following is a version of Theorem 2.1.1.
Proposition A.0.1 (equivalent of Theorem 2.1.1) If htop (f ) < +1, then
(i) convex hull( h0+ ) = p^0,
(ii) ( h0+ )^ = p0 .
106
Remark. Suppose that htop (f ) = +1. Then p0 (t) = +1, t 2 Rd, so
p^0(z) = 1, z 2 Rd. Also H+s (Rd) = htop (f ) = +1; and, from (b) of
Lemma A.0.1, one concludes that convex hull( h0+) is improper (i.e. assumes
the value 1). This implies that ( h0+ )^(t) = +1, t 2 Rd | (ii) is satised.
In contrast, (i) fails | even though convex hull( h0+ )(z) = 1, for z in the
relative interior of the essential domain (cf. Theorem 7.2 [Roc72]); it is +1
or nite outside of this interior.
Proof of Proposition A.0.1. First of all parts (i) and (ii) are equivalent: one
can derive the other by taking convex conjugates of both sides. In particular,
from (ii), after taking convex conjugates, we get (by the involutive property
of ^ | see the page 34) that cl(convex hull( h0+ )) = p^0. Since h0+ is lower
semi-continuous and has bounded essential domain, we can drop \cl" to get
(i) (see Proposition A.0.3). We prove (ii) now.
Clearly ps(0) = H+s = (Rd), and so we get from Lemma A.0.1 the following
(a) ps(0) hs+(z), s; z 2 Rd,
(b) ps(0) supfhs+(z) : z 2 Rdg.
Making the dependence on s explicit (via (A.6) and (A.2)), write (a) as
p0(s) hz; si + h0+(z). Optimizing over all z 2 Rd, we get p0(s) supfhz; si +
h0+(z) : z 2 Rdg = ( h0+ )^(z). Similarly, (b) yields p0 (s) supfhs; zi + h0+ (z) :
z 2 Rdg, i.e. p0(s) h^ 0+(s). Hence, h^ 0+(s) = p0(s), what proves (ii). Q.E.D.
Simple considerations from convex analysis show that, in Proposition
A.0.1, the entropy h0+ can be replaced with h0 . Recall that the set of subdif-
107
ferentials of ps at t 2 Rd is given by
@ps(t) := fz 2 Rd : ps(t0 ) ps(t) + hz; t0 ti for all t0 2 Rdg:
If @ps (t) consists of a single point, then ps is dierentiable at t and @p(t) =
frp(t)g. The following is a convenient characterization (p. 218, [Roc72]):
p^s(z) + ps(t) = hz; ti z 2 @ps(t) t 2 @ p^s (z):
(A:7)
As a corollary, any gradient v := rps(t) is a point of strict convexity of p^s.
(Actually, gradients v are exactly the points at which the epigraph epi(p^s) is
exposed, i.e. there is a supporting hyperplane to epi(p^s) that touches it only at
(v; p^s(v)) | see Corollary 25.1.3, p. 243 in [Roc72].) The following proposition
explains the special role of the gradients in our context.
Proposition A.0.2 If p0 is dierentiable at s 2 Rd and v := rp0 (s), then
h0+(v) = h0 (v) = p^0(v):
(A:8)
By Th. 18.6 p. 167 in [Roc72], the exposed points are dense among all
extremal points. This (using lower semicontinuity of h0 and p^0) leads to the
following corollary strengthening Proposition A.0.1.
Corollary A.0.1 If htop (f ) < +1, then convex hull( h0 ) = p^0 .
Proof of Proposition A.0.2. As we have already said, p^0 is strictly convex at
v, and so h0+(v) = convex hull( h0+)(v) = p^0(v): Now, by (A.7), v = rps(0)
is the unique point v such that
ps(0) = supfhz; 0i p^s(z) : z 2 Rdg = p^s(v):
108
Thus, for any r > 0, maxf p^s+(z) : z 62 Br (v)g < ps(0). However, Lemma
A.0.1 yields
H+s (Rd n Br (v)) maxfhs+(z) : z 62 Br (v)g;
and, since (by Proposition A.0.1) hs+ (z) p^s+ (z), we get
H+s (Rd n Br (v)) < ps(0) = H+s = (Rd):
By (A.4), H s (Br (v)) = H s (Rd) = ps(0), and so hs (v) = p^s (v). Our claim
follows via (A.6) and (A.2).
Q.E.D.
Remark. For readers familiar with [Ell85], it may be worthwhile to note
the following. When p0 (t) is smooth, h0+= (z) = p^0 (z) and Lemma A.0.1
(with s = 0) becomes an analogue of Th. II.6.1 in [Ell85]. Our proof leads
along a dierent path than that of Ellis. We start with hs+= , which are
almost tautologically the right entropy functions (cf. Def. II.3.1, [Ell85]); and
then we show their relation to the free energy ps(t) as a manifestation of the
dual behavior of hs+= and ps(t) under variation of s (see (A.6), (A.2)). This
approach modies in a straightforward way to Ellis's setting yielding what
some may nd to be a simpler and more natural proof of Th. II.6.1.
The following example shows that, in general, hs+ 6= hs .
Example (cf. the example in the rst section of Chapter 2). For
(xi )i2Z 2 f1; 2; 3; 4; 5; 6gZ, (xk ; :::; xl ) is a mod 3 block if xk ; :::; xl are congruent
modulo 3. We call it maximal if xk 1; xl+1 are both not congruent modulo 3 to
xi , i = k; :::; l. The number l k + 1 is referred to as the length of the block.
Fix an arbitrary unbounded subset of natural numbers N . Consider a subshift
109
(; ), where
= fx 2 f1; 2; 3; 4; 5; 6gZ : x has at most three maximal parity blocks
each either innite or with length in Ng
and shifts to the left: (((xi)i2Z))i := xi+1 , i 2 Z. The space is easily seen
to be compact and invariant. Let the observable : ! R3 be given by
8
>
>
>
>
(1; 0; 0); if x0 = 0 mod 3
>
>
>
>
<
(x) := > (0; 1; 0); if x0 = 1 mod 3 :
>
>
>
>
>
>
>
: (0; 0; 1); if x0 = 2 mod 3
For n 2 3N , one has a (1; n)-separated set S with cardinality 2n and
such that (1=n)n(x) = (1=3; 1=3; 1=3), x 2 S . Indeed, consider (xi ) 2 that
have (x0 ; :::; xn=3 1 ), (xn=3; :::; x2n=3 1 ), (x2n=3 ; :::; xn 1) as mod 3 blocks, where
there are 2n=3 possibilities for each block given a residue class mod 3. Thus,
h0+(1=3; 1=3; 1=3) ln 2. (Actually, we have equality since one can easily see
that htop () = ln 2.) On the other hand, we claim that if N is very sparse,
say N := fk!gk2N, then h0 (1=3; 1=3; 1=3) = 1. Indeed, let n = k! 1.
For x = (xi ) 2 , any maximal mod 3 block contained in (x0 ; :::; xn 1) has
length not exceeding (k 1)!. Thus at least one of the three coordinates of
(1=n)n(x) does not exceed (k 1)!=n 1=k. This makes (1=n)n(x) lie in
a denite distance from (1=3; 1=3; 1=3) for all x 2 and innitely many n of
the form n = k! 1, k 2 N. Our claim follows.
110
We nish with the following fact from convex analysis that we used numerous times.
Proposition A.0.3 If f : Rd ! ( 1; +1] is lower semi-continuous and
dom(f ) is bounded, then convex hull(f ) is lower semi-continuous.
Proof. By lower semi-continuity, f has a nite inmum M 2 R. By considering
f M in place of f , we may assume that f is non-negative.
Let K := epi(f ) Rd+1. The essential domain dom(f ) is the projection
of K on the Rd of the arguments. It is enough to demonstrate that conv(K )
is closed, because then it is the epigraph of convex hull(f ) (see the denitions
on p. 36 in [Roc72]).
Suppose that (zn; yn) 2 conv(K ) and limn!1(zn; yn) = (z; y). We want
to show that (z; y) 2 conv(K ). From the Caratheodory theorem ([Roc72]),
+1 = 1 such that
there are (wn;i; tn;i) 2 K and n;i; n > 0, Pdi=1
n;i
(zn; yn) =
dX
+1
i=1
n;i (wn;i; tn;i) + (0; n):
(A:9)
Let B := fi : (tn;i)n2N is unboundedg. Passing perhaps to a subsequence, we
can assume that, as n ! 1, we have tn;i ! ti 2 R, i 62 B ; and additionally,
zn ! z 2 Rd, n ! 2 R, wn;i ! wi 2 Rd and n;i ! i, i = 1; :::n. Note
that there must be i = 0 for i 2 B ; otherwise yn's could not stay bounded.
Thus the sum on the right side below
(z0 ; y0) :=
X
i62B
i (wi; ti) + (0; );
111
is a convex combination, and z0 = z. Also, y0 y because tn;i 0. Since K
is closed, we have (wi; ti) 2 K , so (z0 ; y0) 2 conv(K ). It follows that (z; y) is
also contained in conv(K ).
Q.E.D.
Bibliography
[Abi80]
W. Abiko. The Real Analytic Theory of Teichmuller Space.
Springer-Verlag, Berlin, 1980.
[Alh66]
L. Alhfors. Lectures on Quasiconformal Mappings. van Nostrand, New York, 1966.
[ALM93]
L. Alseda, J. Llibre, and M. Misiurewicz. Combinatorial Dynamics and Entropy in Dimension One, volume 5 of Advanced series
in Nonlinear Dynamics. World Scientic, Singapore, 1993.
[Ang90]
S.B. Angenent. Monotone recurrence relations, their Birkho
orbits and topological entropy. Erg. Th. & Dyn. Sys., 10:15{41,
1990.
[Ban88]
V. Bangert. Mather Sets for Twist Maps and Geodesics on Tori.
volume 1 of Dynamics Reported. Oxford University Press, Oxford, 1988.
[BH92]
M. Bestvina and M. Handel. Train tracks for surface homeomorphisms. CUNY preprint, 1992.
112
113
[Bow71]
R. Bowen. Entropy for group endomorphisms and homogenous
spaces. Trans. Amer. Math. Soc., 153:401{414, 1971.
[Bow75]
R. Bowen. Equilibrium States and the Ergodic Theory of Anosov
Dieomorphism, volume 470 of Springer Lecture Notes in Math.
Springer-Verlag, Berlin, 1975.
[Bow78a]
R. Bowen. Entropy and the fundamental group. In The Structure
of Attractors in Dynamical Systems, volume 668 of Lecture Notes
in Mathematics. Springer, Berlin, 1978.
[Bow78b]
R. Bowen. Markov partitions are not smooth. Proc. Amer. Math.
Soc., 71:130{132, 1978.
[Boy92]
P. Boyland. The rotation set as a dynamical invariant. In Proceedings of IMA Workshop on Twist Maps, volume 44 of IMA
Volumes in Math. Appl. Springer, Berlin, 1992.
[Boy94]
P. Boyland. Topological methods in surface dynamics. Topology
and its Applications, 58:223{298, 1994.
[Boy10]
P. Boyland. Isotopy stability of dynamics on surfaces. IMS,
Stony Brook preprint, 1993/10.
[BW93]
M. Barge and R. Walker. Periodic point free maps of tori which
have rotation sets with interior. Nonlinearity, 6 (3):481{489,
1993.
114
[CSUZ87]
A.A. Chernikov, R.Z. Sagdeev, D.A. Usikov, and G.M. Zaslavsky.
The Hamiltonian method for quasicrtystal symmetry. Physics
Letters A, 125 (2):102{106, 1987.
[Ell85]
R. S. Ellis. Entropy, Large Deviations, and Statistical Mechanics.
Springer-Verlag, New York, 1985.
[FLP79]
A. Fathi, F. Lauderbach, and V. Poenaru. Travaux de Thurston
sur les surfaces, volume 66-67 of Asterisque. 1979.
[FM90]
J. Franks and M. Misiurewicz. Rotation sets of toral ows. Proc.
Amer. Math. Soc., 109 no. 1:243{249, 1990.
[Fra88]
J. Franks. Recurrence and xed points of surface homeomorphisms. Erg. Th. & Dyn. Sys., 8:99{107, 1988.
[Fra89]
J. Franks. Realizing rotation vectors for torus homeomorphisms.
Trans. Amer. Math. Soc., 311:107{115, 1989.
[Fra94]
J. Franks. Rotation vectors and xed points of area preserving
surface dieomorphisms. Preprint, 1994.
[Fri82a]
D. Fried. Flow equivalence, hyperbolic systems and a new zeta
function for ows. Comment. Math. Helv., 57:237{259, 1982.
[Fri82b]
D. Fried. The geometry of cross sections to ows. Topology,
21:353{371, 1982.
[Gro87]
M. Gromov. On the entropy of holomorphic maps. Preprint,
1987.
115
[Han85]
M. Handel. Global shadowing of pseudo-anosov homeomorphisms. Erg. Th. & Dyn. Sys., 5:373{377, 1985.
[Kwa]
J. Kwapisz. A toral dieomorphism with a non-polygonal rotation set. IMS, Stony Brook, preprint #1994/7 (to appear in
Nonlinearity).
[Kwa92]
J. Kwapisz. Every convex rational polygon is a rotation set. Erg.
Th. & Dyn. Sys., 12:333{339, 1992.
[Kwa93]
J. Kwapisz. An estimate of entropy for toroidal chaos. Erg. Th.
& Dyn. Sys., 13:123{129, 1993.
[LL93]
A.J. Lichtenberg and M.A. Lieberman. Regular and Chaotic Dynamics. Springer, Berlin, 1993.
[LM91]
J. Llibre and R. MacKay. Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. Erg. Th. & Dyn.
Sys., 11:115{128, 1991.
[Man87]
R. Mane. Ergodic Theory and Dierentiable Dynamics.
Springer, Berlin, 1987.
[May91]
D. H. Mayer. Continued fractions and related transformations.
In Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces.
Oxford University Press, Oxford, 1991.
[MT91]
B. Marcus and S. Tuncel. The weight-per-symbol polytope and
scaolds of invariants associated with markov chains. Erg. Th.
116
& Dyn. Sys., 11:129{180, 1991.
[MT92]
M. Misiurewicz and J. Tolosa. Entropy of snakes and the restricted variational principle. Erg. Th. & Dyn. Sys., 12:791{802,
1992.
[MZ89]
M. Misiurewicz and K. Ziemian. Rotation sets for maps of tori.
J. London Math. Soc., 40:490{506, 1989.
[MZ91]
M. Misiurewicz and K. Ziemian. Rotation sets and ergodic measures for torus homeomorphisms. Fund. Math., 137:45{52, 1991.
[New89]
S. Newhouse. Continuity properties of entropy. Ann. of Math.
(2), 129:215{235, 1989.
[NPT83]
S. Newhouse, J. Palis, and F. Takens. Bifurcations and stability
of families of dieomorphisms. Publ. Math. IHES, 57:5{71, 1983.
[PP90]
W. Parry and M. Pollicott. Zeta functions and the periodic orbit
structure of hyperbolic dynamics, volume 187-188 of Asterisque.
1990.
[Roc72]
R. T. Rockafellar. Convex Analysis. Princeton Mathematical
Series. Princeton University Press, Princeton, 1972.
[Rue78]
D. Ruelle. Thermodynamical formalism. Encyclopedia of Mathematics and its Applications. Addison-Wesley, London, 1978.
[Thu88]
W. Thurston. On the geometry and dynamics of dieomorphisms
of surfaces. Bull. Amer. Math. Soc., 19:417{431, 1988.
117
[Wal82]
P. Walters. An Introduction to Ergodic Theory, volume 79 of
Graduate texts in mathematics. Springer-Verlag, New York,
1982.
[Zie95]
K. Ziemian. Rotation sets for subshifts of nite type. Fund.
Math., 146:189{201, 1995.
[ZZRSC86a] G.M. Zaslavsky, Mu.YU Zakharov, D.A. Usikov R.Z. Sagdeev,
and A.A. Chernikov. Generation of ordered structures with a
symmetry axis from a Hamiltonian dynamics. Pis'ma Zh. Eksp.
Teor. Fiz., 44 (7):451{456, 1986.
[ZZRSC86b] G.M. Zaslavsky, Mu.YU Zakharov, D.A. Usikov R.Z. Sagdeev,
and A.A. Chernikov. Stochastic web and diusion of particles in
a magnetic eld. Sov. Phys. JETP, 64:294{303, 1986.
Index
(; n)-separated set, 29
points of strict convexity, 34
(v)
htop
(f; ), 30
(v)
htop
(F ), 69
primitive periodic orbit, 4
proper convex function, 34
relative interior, 34
pseudo-Anosov rel a nite set, 14
ane hull, 34
rotation set, 2
rotation vector, 2
connecting length, 46
scaold of subshifts of nite type, 62
convex hull, 34
critical points, 96
support of v in (; ), 62
displacement, 2
tent function K (v ), 69
eective domain, 34
topological entropy at the asymptotic
the set of asymptotic averages, 29
equilibrium state, 38
average, 30
Gibbs weight, 102
topological entropy at the rotation vec-
involutive property of convex conju-
transfer matrix, 48
tor, 69
gation, 34
transition graph, 45
Minkowski sum, A+B, 55
observable, 29
118