Combinatorics of Torus Dieomorphisms
Jaroslaw Kwapisz
Department of Mathematical Sciences
Montana State University
Bozeman MT 59717-2400
tel: (406) 994 5358
fax: (406) 994 1789
e-mail: jarek@math.montana.edu
web page: http://www.math.montana.edu/~jarek/
May 28, 2002
Abstract
We construct dynamical partitions of the torus for a dieomorphism that
is isotopic to the identity. The existence and the combinatorics of the partitions is soleley deteremined by the rotation set of the dieomorphism. When
the rotation set consists of a single non-resonant vector, there is a whole hierarchy of partitions analogous to the partitions of the circle into the closest
return intervals under an irrational circle rotation. In particular, all such
torus maps are innitely renormalizable in a natural sense.
1 Introduction
Taking Td to be the d-dimensional torus obtained as the quotient R d =Zd where the
integer lattice Zd R d acts by translations, let Di0(Td ) be the space of all C 1dieomorphisms f : Td ! Td that are C 1 -isotopic to the identity and let Di0 (R d )
be the space of all their lifts to R d . For F 2 Di0(R d ), the rotation vector of F is
dened as
F n ( p) p
F := nlim
!1
n
provided the limit exists and is independent on p 2 R d . If F fails to exist, one
speaks of the rotation
set (F ) of F dened as the set of all limit points of sequences
n (pk ) pk
F
of the form nk where nk ! 1 and pk 2 R d (see [17]).
1
In connection with the classical problem of stability and breakdown of quasiperiodic invariant tori, of particular interest are the non-resonant torus maps, that
is f 2 Di0 (Td ) with a lift F 2 Di0(R d ) such that F = (r1; : : : ; rd) is well
dened and
d
X
i=1
!
iri = 0 ; i 2 Z ) (i = 0 for i = 0; 1; : : : ; d) :
The fundamental question is that about dynamical classication of such maps;
in particular, about existence of a conjugacy or a semiconjugacy to the simplest
model provided by the quasi-periodic translation, T : p 7! p + , = F . In
dimension d = 1, the resulting theory of circle dieomorphisms with irrational
rotation number is rather complete and, in a nutshell, amounts to the following
[4]:
(i) (Poincare) A semiconjgacy, h : T ! T, T h = h f , always exists.
(ii) (Denjoy) h is a conjugacy if f is suciently regular (f 0 is BV).
(iii) (Arnold, Herman, Yoccoz, Katznelson and Ornstein : : : ) The smoothness of
h is determined by the smoothness of f and the number theoretical properties
of F .
The case of dimension d = 2 and higher is much harder: the skew-product examples
of Herman in [8, 9] (see also [7, 1]) show that the analogues of all three assertions
above generally fail. (In particular, from 4.5 of [9] one can extract real-analytic
f 2 Di0 (T2 ) with Diophantine F that are not even semi-conjugated to the
translation.) Beyond the local situation when f is near translation and some sort
of KAM technique applies, little is known in positive (c.f. [10]). The failure of
(i) is the main culprit depriving one on the outset of the topological model on
which the analytical arguments could be layered1 . However, the success of (ii)
and (iii) in the circle case depends not as much on (i) per se as on its corollary
about the existence of the hierarchy of dynamical partitions of T associated to the
process of Diophantine approximation of (see e.g. (1.6) page 26 in [4]). It is our
main result that the analogous partitions exist in dimension d = 2 and that one
can justiably say that all non-resonant torus maps with a given rotation vector combinatorially look like the translation T. Thus we resolve the problem of nding
a suitable generalization of (i) to d = 2 and oer a departure point for attempts
to generalize (ii) and (iii).
To state the result, we rst extend the notion of Farey neighboring fractions.
Denition 1.1 Suppose that we are given three rational vectors 1 , 2, and 3
written in the lowest terms 1 = (u1 =u3; u2 =u3), 2 = (v1 =v3 ; v2 =v3 ), and 3 =
1 Assuming (i) one can make some progress see [19, 18].
2
(w1=w3; w2=w3) with u3; v3 ; w3 > 0. The vectors (i )i=1;2;3, form a Farey triple i
2
3
u1 v1 w1
det 4u2 v2 w25 = 1:
u3 v3 w3
Given a subset B R 2 , (i )i=1;2;3 is a Farey triple for B i it is a Farey triple and
B is contained in the open simplex with vertices at 1, 2 , and 3. Moreover, if
B = fg is a single point, we simply speak of (i )i=1;2;3 as a Farey triple for .
Theorem 1.2 Suppose that f 2 Di0(T2 ) and its lift F 2 Di0 (R 2 ) are given
together with 1 := (u1 =u3; u2 =u3 ), 2 = (v1 =v3 ; v2 =v3 ), and 3 := (w1=w3 ; w2 =w3 )
such that (i)i=1;2;3 is a Farey triple for (F ), then there exist three smooth quadrilaterals (i.e. dieomorphic images of closed Euclidean rectangles) J0 , K0 , and L0
in T2 such that, denoting Ji := f i(J0 ), Ki := f i(K0), and Li := f i (L0 ), the family
fJ0; : : : ; Jw 1; K0; : : : ; Ku 1; L0; : : : ; Lv 1g;
is an essentially disjoint covering of T2 (i.e. it covers all of T2 and its elements
3
3
3
have pairwise disjoint interiors). The combinatorics of this covering is independent
of f in the sense that, if f and f are two maps that satisfy the above hypotheses,
then there exists a homotopic to the identity homeomorphism h : T2 ! T2 that
maps the covering for f to the one for f, and the following diagram commutes
T2 n (Jw 1 [ ?Ku 1 [ Lv 1)
f
T2 n (Jw 1 [ K u 1 [ L v 1)
f
3
3
3
?
hy
3
3
3
! T2 n (J0 [?K0 [ L0)
?
hy
:
(1.1)
! T2 n (J0 [ K 0 [ L 0)
Moreover, if f is a translation, J0 , K0 , and L0 can be obtained as projections to
T2 of three Euclidean parallelograms in R 2 every two of which share a side (as in
Figure 4.3).
We shall refer to the covering in the theorem as a dynamical tiling for f . Figure 1.1
depicts example tilings for the translation F = T where = (; 1= 1) with
3 2 1 = 0 and for the area preserving version of the standard map
F = Fa;b; = T Vb Ha where Ha(x; y) = (x + 2a sin(2y); y) with a = 0:45 and
Vb(x; y) = (x; y + 2b sin(2x)) with b = 1:8. Theorem 1.2 says that the two pictures
are homeomorphic and that f and f are conjugated save for three tiles. This of
course yields no immediate conclusions about the asymptotic dynamics of f (in a
way consistent with existing counterexamples). On the other hand, for any nonresonant f 2 Di0 (T2 ), there is a sequence of Farey triples ((in) )3i=1, n 2 N , such
that limn!1 maxi j (in) j = 0, := F , and diagram (1.1) supplies a sequence hn
of dieomorphisms that conjugate f to the translation onto a progressively bigger
3
Figure 1.1: Dynamical tilings for a standard map and the translation with the same rotation vector associated to u = (927; 778; 504), v = (504; 423; 274), and w = (274; 230; 149)
subset of T2 so that limn!1 maxp2T dist(f hn(p); hn f (p)) = 0. Passage to the
limit limn!1 hn to obtain a conjugacy or a semiconjugacy is not possible in general
without further control of the geometry of Ji(n) 's. (It is only for d = 1 that the
order structure on T readily secures Poincare's semiconjugacy.)
Theorem 1.2 is more than an approximation result: it is a result on existence
of renormalizations for f . Indeed, we shall show in Section 4 that each of the
tiles J0 , K0 , and L0 has a natural identication on the boundary making it into
a torus, and that the return maps J0 ! J0, K0 ! K0, and L0 ! L0 yield
again dieomorphisms isotopic to the identity. In particular, the return map to
J0 [ K0 [ L0 = Jw [ Ku [ Lv yields a map in Di0(T2 ) that can be rightfully
called a (u; v; w)-renormalization of f (after rescaling its domain back to T2 ). This
is inspired and in perfect harmony with the circle case where the renormalization arises from the return map to a pair of suitable adjacent segments (see e.g.
[4]). Following the established paradigm, convergence of the renormalizations (of
which a whole sequence exists for non-resonant f ) may lead to conjugacy results;
although, implementation of this idea will be anything but easy and is left for
future. We should not try to leave the impression that we are the rst to attempt
renormalization involving invariant two-tori | see e.g. [11, 14, 12]. At this point,
we lack the expertise to compare these aproaches beyond noticing that ours is more
geometrical and global (i.e. not conned to near translations) but is restricted (for
now) to the context of a single torus.
The formulation of Theorem 1.2 by necessity omits many key aspects of the
theory which emerge in all clarity only from the proof. For a quick overview,
it may be worth skimming Section 2,3, and 4. Section 2 establishes the line of
2
3
3
3
4
the argument and contains the key geometrical ideas. Section 3 provides detailed
description of the combinatorics of the dynamical tiling by dening and analyzing
their simplest models in the form of the stepped translation on the stepped plane and
the stepped torus. Here we borrow and extend the terminology that we learned from
R.F. Williams, who independently studied certain stepped planes in connection with
DA tiling spaces (see [21]). Section 4 builds on Section 3 to oer the beginning
of the renormalization theory while a more systematic development is relegated to
[16]. Finally, Section 5 contains the most technical topological part of the proof of
Theorem 1.2. Unlike Sections 2,3, and 4, which are easily translated to the context
of d > 2, these arguments are intimately connected with the topology of twodimensional submanifolds of T3 and their generalization to d > 2 is not obvious.
We should also mention that this work continues and to some extent overlaps [15].
Reading the introduction to [15] may provide further perspective on our results. In
particular, our Theorem 1.2 should be compared with Translation Loop Theorem
(Theorem 1.3) in [15]. Finally, let us stress that our main objective in the present
paper was to present the key ideas with a minimal burden of technicalities. For
instance, our assumption about C 1-smoothness can almost surely be weakened but
we adopted it to be able to invoke Fried's results on existence of cross sections.
Note added in revision: Robert MacKay pointed out that renormalizations on
the n-dimensional torus were studied in [20]. The emphasis of [20] is on elucidating the combinatorial structure of the rigid translations by coding orbits via \even
n-colourings of the integers". The coding domains are obtained via a staircase construction which appears to coincide with the stepped plane we use. In particular,
our Section 3 and Theorem 2.3 nd their counterpart in Theorem 1 of [20]. Coincidentally, similar ideas in various guises appear also in the study of the generalized
Sturmian sequences and the associated tiling spaces by by G. Rauzy, P. Arnoux,
S. Ito and M. Ohtsuki, V.Berthe and L. Vuillion and others | see Chapters 7
and 8 in [5] for more information and references. What sets our work apart is
that we uncover combinatorial structure of maps that are not rigid translations |
nor are assumed to be conjugated or Denjoy-semi-conjugated to such maps as in
Theorem 2 of [20].
Let us also briey comment on the interplay between renormalization and owequivalence. It is known that the two eect the same scaling of the frequencies
of motions and therefore coincide up to conjugacy. However, the connection is
typically formalized (see e.g. Prop. 1.1 in [13]) in terms of Zd+1 actions on R d (or
commuting d + 1-tuples of maps), which obscures the existence of the dynamical
partitions we construct.
Acknowledgment: The author would like to thank Phil Boyland, Robert
MacKay, and Hans Koch for their comments on the manuscript.
5
2 Outline of Proof
Let us outline the strategy of the proof of Theorem 1.2 while relegating detailed
arguments to the subsequent sections.
Suppose that the hypothesis of Theorem 1.2 are satised. As a rst step, we
use the fact that f is in fact smoothly isotopic to the identity to suspend f into
a C 1-ow : R T3 ! T3 . Here, we think of the universal cover R 2 of T2 as
the (x; y)-plane S~ := fz = 0g embedded in the (x; y; z)-space R 3 and of T2 as
the corresponding two torus S := T (S~), where T : R 3 ! T3 is the canonical
projection. The suspended ow induces f as the return map to S . The (unique)
lifted ow ~ : RR 3 ! R 3 has ow lines crossing S~ transversally with z-coordinate
changing the sign from to +, and F : S~ ! S~ given by
F = T(0;10;1) ~1jS~
(2.1)
is a lift of f . We shall assume that this is the original F satisfying the hypothesis
of the theorem (as this can be always achieved by adjusting the initial isotopy used
to construct the suspension).
Let A 2 SL3(Z) be the matrix
2
3
u1 v1 w1
A := 4u2 v2 w25 :
(2.2)
u3 v3 w3
The hypothesis on the rotation set of f guarantees that the set of homological directions of | as dened in [6] | is contained in the cone AR 3+ , where R + = (0; 1).
Fried's theorem (Theorem D in [6]) supplies therefore global cross-sections J , K ,
and L that are 2-tori homologous to the ane tori T (lin(u; v)), T (lin(v; w)),
T (lin(w; u)) | respectively. (How Fried's theory specializes to the context of
torus maps is explained in some more detail in Section 2 of [15].)
The cross-sections may, a priori, intersect inside T3 in a very complicated manner. The heart of our argument lies in assuring that J , K , and L form a clean
triple of cross-sections in the sense of the following denition (c.f. Denition 1.1 in
[15]).
Denition 2.1 A triple of C 1 -embedded tori J , K , and L in T3 is called clean if
J \ K , K \ L, and L \ J are simple closed loops spanning the homology H1(T3 ),
J \ K \ L is a single point, and all the above intersections are transversal. If
additionally J , K , and L are cross-sections to a ow, then J , K , and L are called
a clean triple of cross-sections.
This roughly says that tori forming a clean triple have the simplest possible topology of their mutual intersections permitted by their cohomology classes (as exemplied by the ane tori). The key theorem below is shown in Section 5 and
it extends Theorem 3.6 in [15] dealing with the much simpler case of two cross
sections.
3
3
3
3
6
3
Theorem 2.2 (Cleaning) Suppose that : T3 R ! T3 is a C 1 -smooth ow on
T3 . If there are three (global) cross sections J , K , and L whose cohomology classes
form a basis of the rst cohomology of T3 over Z, such cross sections exist that
form a clean triple of cross sections. Moreover, the clean triple can be obtained by
modifying each of J , K , and L via an isotopy in T3 .
It is left to derive all the assertions of Theorem 1.2 from the existence of a clean
triple of cross sections. To achieve a clear geometrical picture, let us conjugate (and its lift ~) by a dieomorphism isotopic to the identity so that the three clean
cross sections J , K , and L are straightened out to the ane tori obtained as the
quotients (under Z3) of the planes
J~ := lin(u; v); K~ := lin(v; w); L~ := lin(w; u):
This step, formalized in Lemma 6.2, is quite intuitive since one expects that cutting
T3 along J , K , and L yields a 3-ball that can be mapped to [0; 1]3 so that the
natural boundary identications transform into those induced by Z3.
Let us retain the old notation after the conjugation. Note that, typically, S is no
longer an ane torus nor is S~ a plane anymore. The ow ~ (after the conjugacy)
is transversal to the planes J~, K~ , and L~ . Those planes bound the cone AR 3+ ,
which is carried by the ow into itself (see Figure 2.1). The rest of the argument is
based on the very simple idea that the ow lines on T3 , having their lifts trapped
in AR 3+ , are approximately determined by the way AR 3+ projects to T3 ; and the
approximation is better if the cone AR 3+ is narrow | which happens whenever
ui, vi , and wi are large. More precisely, for p 2 R 3 , the ow line ~R(p) intersects
each of S~, J~, K~ , and L~ exactly once so that the three faces of the fundamental
parallelepiped
P := fu + v + w : 0 ; ; 1g
that meet at the vertex (0; 0; 0) | see Figure 2.1 | map dieomorphically, by the
holonomy along the ow lines, to three quadrilaterals in S~:
J~0 := fp 2 S~ : ~R(p) \ J~ \ P 6= ;g;
K~ 0 := fp 2 S~ : ~R(p) \ K~ \ P 6= ;g;
L~ 0 := fp 2 S~ : ~R(p) \ L~ \ P 6= ;g:
For i 2 N , let us set
J~i := F i(J~0);
K~ i := F i(K~ 0);
L~ i := F i(L~ 0 );
and write Ji, Ki, and Li for the corresponding projections to S . In order to
complete the proof of Theorem 1.2, we shall prove in the next section the following
more precise result.
7
u
P
S~
v
w
L~ 0
K~ 0
J~0
Figure 2.1: The Cone AR3+ and the ow (after straightening the cross-sections).
Theorem 2.3 (Combinatorial Part of Theorem 1.2) The family of quadri-
laterals
Ff;A := fJ0; : : : ; Jw 1 ; K0; : : : ; Ku 1 ; L0; : : : ; Lv 1g;
3
3
3
obtained as above from a clean triple of cross sections to a suspension ow of
f 2 Di0(T2 ), is an essentially disjoint covering of T2 that satises the assertions
of Theorem 1.2.
We note that, unlike Theorem 1.2, Theorem 2.3 and its proof generalize to dimensions d > 2 at a mere cost of complicating the notations.
3 The Stepped Translation
This section explores the link between clean triples of cross-sections and dynamical
tilings and culminates in a proof of Theorem 2.3. The stepped translation will be a
special map serving as a combinatorial model for all f 's satisfying the assumptions
of Theorem 2.3.
For convenience, besides the straightening conjugacy performed in Section 2, we
further conjugate the ow via the linear map induced by the matrix A 1 , which
8
carries (u; v; w) onto the standard basis (e1 ; e2; e3 ). Thus, in the place of ~t , we
consider
A 1 ~t A:
As before, we shall retain the pre-conjugacy notations. Observe that the the crosssection S (over which the suspension was built) is now cohomologous to the ane
torus
^ := fdz A = u3dx + v3dy + w3dz = 0g;
and the lift S~ is an embedded topological plane in R 3 that is Z3-equivariantly
homotopic to
~ := fu3x + v3y + w3 z = 0g:
The time translation, formerly T(0;0;1) , is represented now by TA e ; and (2.1)
becomes
F = TA 1 e ~1jS~:
(3.1)
Our strategy is to exploit that the return maps to cohomologous cross sections
are conjugated and replace S by a section that comes with a natural dynamical
tiling. For a technical reason | the new section being degenerate in T3 | we choose
to work exclusively with lifts. In particular, we shall need the cyclic (suspension)
covering of ,
^ : R 3 = R ! R 3 =
where the sublattice Z3 is that of the spatial translations:
:= ZA 1e1 + ZA 1e2 :
(3.2)
Having xed any connected lift S^ of S to R 3 =, we see that
f^ = T^A 1 e ^1 jS^;
(3.3)
1
1
1
3
3
3
were T^v denotes the quotient by of Tv : R 3 ! R 3 , is conjugated to f : S ! S
via the canonical projection from R 3 = to T3 .
The hyperplane ~ cuts R 3 into two open half spaces: ~ and ~ +, where ~
is the half space containing A 1 e3, i.e. the one exited by the ow through ~ .
Referring to any Z3-translate of [0; 1]3 as a lattice cube, we dene H to be the
union of all closed lattice cubes with their interiors entirely contained in ~ ,
[
H := fQ a lattice cube : int(Q) ~ g:
Also, let ~ be the boundary of H ,
~ := @H :
Note that ~ and ~ are invariant under so that we have a well dened quotient
^ := ~ =:
9
Denition 3.1 With any A = (u; v; w) 2 SL3 (Z) we associate the following objects:
~ called the stepped plane,
^ called the stepped torus,
T := T A e j~ : ~ ! R 3 called the stepped plane translation,
the quotient of T , T^ : ^ ! R 3 =, called the stepped torus translation.
1
3
We shall see that ~ and ^ are indeed topologically a plane and a torus but neither
~ ~ nor T^ ()
^ .
^ As we already mentioned in the introduction, stepped
T ()
planes can be found in [20] and in the literature on tiling spaces (see e.g. [21, 5]).
One way to understand ~ (and progress towards the proof of Theorem 2.3)
is by analyzing the family Q^ of all the lattice cubes in H = that meet ^ and
~
the family of all their lifts Q~ , which are the lattice cubes in H that meet .
~
Figure 3.1 is a 2-d projection of an example Q. Note that the six faces of any
lattice cube Q split into two families: the three entrance faces sharing the vertex
min Q := (minQ x; minQ y; minQ z) through which the ow enters Q, and the three
exit faces sharing the vertex max Q through which the ow exits Q. (In Figure 3.1,
only the exit faces are visible | c.f. also Figure 4.2.) It is easy to see that ~ does
not meet the entrance faces of Q 2 Q~ : the negative cone, max Q R 3+ , is disjoint
from ~ , so a cube Q0 neighboring with Q across an entrance face D cannot hit ~ ,
~
which means that D is not contained in .
Fact 3.2 ~ constitutes a cross-section to ~ in the sense that every ow line meets
~ exactly once (as it exits H ), and ~ is a topological plane homeomorphic to S~
via the natural holonomy
h~ : ~ ! S~
that sends p 2 ~ to the unique point h~ (p) 2 S~ on the ow line ~R (p).
Proof: Because ~ is a lift of a suspension ow, given p 2 R 3 , there are t < t+
such that ~( 1;t )(p) ~ and ~(t ;1)(p) ~ + , and limt!1 dist(~t(p); ~ ) = 1.
Thus ~R(p) hits both H and its complement, and we conclude that ~ = @H
meets all the ow lines. That each ow line intersects ~ exactly once follows from
the fact that ~ (exclusively made of the exit faces of Q~ ) is topologically transverse
to the ow with all ow lines exiting H through ~ . Hence, we have shown that
~ is a cross-section. The assertion about the holonomy follows readily. 2
We turn now our attention to the natural tiling of ~ into the three families of
cubic faces, each collecting the faces Z3-congruent to the unit square in one of the
^ we denote by J^, K^ , and L^
planes fz = 0g, fy = 0g, or fx = 0g. Descending to ,
^
the three corresponding families that tile (see Figure 3.1).
Fact 3.3 Writing #A for the cardinality of a set A, we have
+
10
10
9
8
7
0
1
2
3
12
7
11
6
5
4
0
1
11
10
9
8
5
4
3
2
9
8
7
6
1
12
12
6
5
0
1
2
11
5
10
4
9
3
0
Figure 3.1: The Stepped Plane ~ for A = [u; v; w] with u = (24; 20; 13), v = (13; 11; 7);
and w = (7; 6; 4) as viewed from the point (2000; 1000; 1000). (For the right perspective
think about the marked vertices as sticking out.) The dashed lines outline a fundamental
parallelogram in the plane ~ of the sublattice . The only (mod ) cube Q0 that meets
~ is white; all other cubes are just behind ~ and the shade darkens with increasing
~ of which only
distance from ~ . Modulo , there are 13 + 7 + 4 cubes touching ,
13 = maxf13; 7; 4g are showing a face. The number k (by the max vertex) identies a
cube as a lift of T^ k Q0 , where T^ = T A e = T(1;4; 4) T(1; 2;0) (mod ) is the stepped
torus translation. The ow (not pictured) pierces ~ transversally from behind and moves
roughly toward the viewer. If the perspective is ignored, the two dimensional picture
shows the dynamical tiling for the translation by (68=37; 57=37), the suspension of which
is the constant ow in the direction (2; 1; 1).
1
3
11
(i) #J^ = w3 , #K^ = u3 , and #L^ = v3 .
(ii) #Q^ = u3 + v3 + w3 .
(iii) T^ (Q~ ) H and #(Q^ n T^ (Q^ )) = 1.
Proof: Fix the orientation on ~ determined by the (outward) normal
(u3; v3; w3).
(i) Consider the projection to R 3 = of the constant vector eld everywhere
equal to e3. The ux through ^ is clearly equal to #J^. On the other hand, the
fundamental parallelogram for the torus ^ is spanned by fA 1e1 ; A 1e2 g and the
cross-product A 1e1 A 1e2 = (u3; v3 ; w3), so the ux through ^ is equal to the
Euclidean scalar product h(u3; v3; w3); e3i = w3. The divergence being zero, the
two uxes are equal by the Stokes Theorem. Thus #J^ = w3. The analogous
arguments show the two other equalities.
(ii) Consider on R 3 = the quotient of the constant vector eld ( 1; 1; 1).
The ux through ^ (and any of its translates) is the same as the ux through ^
and thus equals h(u3; v3; w3); ( 1; 1; 1)i = u3 v3 w3.
At the same time, since all the faces in ~ are exit faces, T( 1; 1; 1) maps H
strictly into itself and therefore Q^ consists exactly of the lattice cubes between ^
^ Thus #Q^ can be computed as the volume trapped between ^
and T^( 1; 1; 1) ().
^
and T( 1; 1; 1) ():
#Q^ =
Z 1
[The
0
ux through
^ dt =
Tt( 1; 1; 1) ()]
(u3 + v3 + w3 ) 1
where we used the classical interpretation of \ux" known as the Reynolds' Transport Theorem (see (6.1) in Theorem 6.1 in the appendix).
(iii) That T (Q~ ) H follows immediately from the denitions of H and
Q~ and the fact that T (~ ) ~ . Thus T^ (Q^ ) H =, and the cardinality
^ and .
^ Taking
#(Q^ n T^ (Q^ )) coincides with the volume V trapped between T^ ()
the constant vector eld equal to A 1e3 , we compute the ux through ^ to be 1
(since det A = 1). Therefore,
V=
2
Z 1
0
[The ux through T
^ dt = 1 1:
tA 1 e3 ()]
~ 6 ,
~
Finally, we investigate T to bring out the key fact that, although T ()
T has an interesting action on ~ that leaves ~ nearly invariant. Let Q0 2 Q^ be
^
the cube with max Q0 = (0; 0; 0) (mod ); we call Q0 the fundamental cube of .
Also, let J^0 , K^ 0, and L^ 0 be the exit faces of Q0 | all three are contained in ^ (c.f.
Figure 3.1).
Fact 3.4
12
(i) Q0 is the unique cube in Q^ that intersects ^ ;
(ii) Q^ n T^ (Q^ ) = Q0 ;
(iii) J^ n T^ (J^) = J^0, K^ n T^ (K^ ) = K^ 0 , L^ n T^ (L^) = L^ 0 ;
(iv) Q^ = fQ0 ; T^ (Q0 ); : : : ; T^ u +v +w 1 (Q0 )g;
3
3
3
(v) J^ = fT^ j (J^0 )gwj =0 1 , K^ = fT^ j (K^ 0)guj =0 1 , and L^ = fT^ j (L^ 0 )gjv=01 .
3
3
3
Proof: (i) For Q 2 Q~ , int(Q) \ ~ = ; by denition of H , so that if Q \ ~ 6= ;,
then Q \ ~ contains a vertex (in fact, Q \ ~ = max Q). It follows that the cubes in
Q~ hitting ~ are in 1-1 correspondence with the lattice points on ~ . We are done
by recalling that A 1e1 and A 1 e2 span = Z3 \ ~ so that the origin 0 (mod )
is the sole element of ^ \ Z3=, the only lattice point in ^ .
(ii) By (iii) of Fact 3.3, we know Q^ n T^ (Q^ ) is a single lattice cube. Because
1
A e3 2 ~ +, T 1([ 1; 0]3) 6 ~ . Since [ 1; 0]3 is a lift of Q0 , this implies that
Q0 62 T^ (Q^ ) by the denition of Q^ (and ~ ).
^ into cubic faces. By
(iii) J^ [ K^ [ L^ tiles ^ and T^ (J^ [ K^ [ L^) tiles T^ ()
^
^
^
^ and ^
(ii), there is exactly one cube, Q0, between T () and . Therefore, T^ ()
coincide save for the faces of Q0 . The three exit faces of Q0, J^0, K^ 0; and L^ 0 , belong
^ which leaves the three remaining entrance faces in T^ ()
^ | (iii) follows.
to ,
(iv) Consider Q 2 Q^ . From limn!1 dist(~ ; T n(~ )) = 1, T^ nQ 62 Q^ for some
n > 0; furthermore, we claim that T^ n+1(Q) = Q0 for the minimal such n denoted
n(Q). This is because (ii) asserts that Q0 is the only cube leaving Q^ under T^ 1.
Clearly, (iv) shall follow if n(Q ) = u3 + v3 + w3 for some Q 2 Q^ . Since T^ k (Q) 6=
^ and the map n : Q^ ! f1; : : : ; #Qg
^ is
T^ j (Q) for k 6= j , n(Q) 2 f1; : : : ; #Qg
injective. Consequently, n is surjective, and we can set Q := n 1 (#Q^ ). We are
done by (ii) of Fact 3.3.
(v) By using (iii), the arguments for (iv) adapt word by word. 2
We are ready to give a proof of Theorem 2.3.
Proof of Theorem 2.3: Recall the homeomorphism h~ : ~ ! S~ supplied by Fact 3.2
and let h^ : ^ ! S^ be the quotient by . The curvilinear parallelograms J0 , K0 ,
and L0 (dened in the previous section) are given by
J0 = A h^ (J^0 ); K0 = A h^ (K^ 0 ); L0 = A h^ (L^ 0 )
where A is the initial linear conjugacy performed at the beginning of this section
| which we agreed to suppress, so we shall work with h^ (J^0 ), h^ (K^ 0), and h^ (L^ 0 ).
Set J^i := T^ i(J^0 ), K^ i := T^ i(K^ 0 ), and L^ i := T^ i (L^ 0). From part (v) of Fact 3.4,
fh^ (J^0); : : : ; h^ (J^w 1); h^ (K^ 0 ); : : : ; h^ (K^ u 1); h^ (L^ 0 ); : : : ; h^ (L^ v 1 )g
3
3
3
constitutes an essentially disjoint covering of the torus ^ . Writing (p) for the
^
^ so that ^() = ^T^^ ()
is the holonomy from ^ to
ight time of p 2 ^ to T^ 1 ()
1
13
^ we see that the following diagram commutes
T^ 1(),
^ ^ ^ T ! ^
?
?
()
?
^hy
(3.4)
?
^hy
T^ ^1
S^
! S^
However, f^ = T^ ^1jS^ by (3.3), and () = 0 everywhere except on J^w 1 [ K^ u
L^ v 1 as it follows from (v) of Fact 3.4. Hence,
T^
!
^ n (J^0 [ K^ 0 [ L^ 0 )
^ n (J^w 1 [ K^ u 1 [ L^ v 1)
3
3
1
[
3
3
?
^h?
y
3
3
?
^h?
y
S^ n (h^ (J^w 1 ) [ h^ (K^ u 1) [ h^ (L^ v 1 ))
3
3
f^
! S^ n (h^ (J^0) [ h^ (K^ 0) [ h^ (L^ 0 ))
3
Because f^ is naturally conjugated to f , this last diagram translated to the original
(pre-conjugacy) setting yields
^
^ n (J^w 1 [ K^ u 1 [ L^ v 1) T ! ^ n (J^0 [ K^ 0 [ L^ 0 )
?
?
?
?
(3.5)
y
y
3
3
3
S n (Jw 1 [ Ku 1 [ Lv 1)
3
3
3
f
! S n (J0 [ K0 [ L0 )
The commuting diagram in Theorem 2.3 is obtained by putting together two copies
of the above diagram, one for f and another for f. 2
4 Renormalization
In this section, we make precise our claim from the introduction that Theorem 1.2
can be viewed as a result on existence of renormalization for maps in Di0(T2 ).
The main point is that renormalization is global (i.e. not restricted to perturbations
of the translations). Our considerations parallel those in [16] dealing with a simpler
renormalization suitable for annulus maps and maps in Di0 (T2 ) whose rotation
set is localized only in one direction. More detailed discussion is relegated to [16].
Denition 4.1 Given A 2 SL3 (Z),
3
2
u1 v1 w1
A = 4u2 v2 w25 ;
u3 v3 w3
with u3 ; v3 ; w3 > 0, we say that f 2 Di0 (T2 ) is A-renormalizable i the stepped
torus ^ associated to A can be mapped homeomorphically via some h : ^ ! S = T2
so that, taking Ji := h(J^i), Ki := h(K^ i), and Li := h(L^ i ), the diagram (3.5)
commutes.
14
The idea is that T2 has a tiling that maps under f in the same way as the canonical
tiling of the stepped torus under T^ . (As it will become clear h may well be required
to be piecewise smooth.)
Since J^0 , K^ 0, and L^ 0 are lattice cube faces in R 3 , their opposite sides are naturally identied by Z3; and the homeomorphism h transports those identications
to the topological quadrilaterals J0, K0, and L0.
Fact 4.2 The identications of the opposite sides of J0, K0 , and L0 are eected
by iterates of f and are as depicted in Figure 4.1.
f u +v (0) u
f (J0 \ K0 )
f v (L0 \ J0 )
f u (0)
J0
f v (0)
0 L0 \ J0
J0 \ K0
3
3
3
3
3
3
f v3 ( K
0
f u (K0 \ L0 )
K0 \ L0
\ L0 )
3
L0
K0
f w +u (0)
3
f v +w (0)
f w (J0 \ K0)
3
3
3
3
f w (L0 \ J0)
3
f w (0)
3
Figure 4.1: Identications: the parallel sides map to one another under an iterate of f .
Proof: J^0, K^ 0, and L^ 0 were dened as the exit faces of the fundamental lattice cube Q0 and they are the three faces sharing the vertex 0 (cf. Fact 3.4 and
Figure 3.1). From (ii) and (v) of Fact 3.4, J^w = T^ w (J^0 ), K^ u = T^ u (K^ 0 ), and
L^ v = T^ v (L^ 0 ) are the three entrance faces of the cube Q0 (see Figure 4.2). Since
T^ is a quotient of a translation, the arrangement of the faces in Q0 easily reveals
the iterates of T^ identifying pairs of parallel sides of J^0, K^ 0, and L^ 0 . Diagram 3.5
assures that the same iterates of f identify the opposite sides of J0, K0, and L0 . 2
Let J0= , K0= , and L0= be the tori obtained by identifying the opposite
sides of J0 , K0 , and L0 .
3
3
3
15
3
3
3
J^0
K^ u
0
L^ 0
L^ v
3
K^ 0
0
J^0
3
K^ 0
J^w
3
Figure 4.2: The cube Q0 and the holonomy from the entrance faces to the exit faces.
Proposition 4.3 Suppose that f 2 Di0(T2 ) is A-renormalizable. Then the sus-
pension ow of f has a well dened cross-section J in the cohomology class represented by lin(u; v) (mod Z3); and the map RJ (f ) : J0 = ! J0 = induced on the
torus J0 = by the rst return to J0 under f is conjugated to the Poincare return
map J : J ! J for the ow. The conjugacy is via a map isotopic to the identity.
The analogous statements are also true for K0 and L0 .
0
Observe that the existence of the cross sections in the cohomology classes of
lin(u; v), lin(v; w), and lin(w; u) (mod Z3) guarantees (see Section 2 in [15]) that
the set of homological directions of the ow is contained in the cone spanned
by fu; v; wg, which implies that 1 = (u1=u3; u2=u3), 2 = (v1 =v3; v2 =v3), and
3 = (w1=w3; w2=w3) form a Farey triple for the rotation set of f . This complements Theorem 2.3 and yields the following characterization of A-renormalizability.
Corollary 4.4 A mapping f 2 Di0(T2 ) is A-renormalizable i (1 ; 2; 3 ), dened above, is a Farey triple for (F ) for some lift F 2 Di0 (R 2 ) of f .
Proof of Proposition 4.3. The plan is to rst construct the conjugacy under
the additional hypothesis that J0, K0 , and L0 are obtained from a clean triple of
cross sections (J; K; L) via the construction described in Section 3. In the second
part of the proof we shall show that the hypothesis can be satised when f is an
A-renormalizable map.
Thus we rst assume that T2 is embedded as S^ in the cyclic covering of the
suspension ow of f (i.e. R 3 = ' T2 R , see (3.2)) and that h^ : ^ ! T2 = S^ is
given by the holonomy between S^ and the stepped torus ^ . The three clean cross
sections are just the quotients to T3 of J^0 , K^ 0 , and L^ 0 . Denoting by fA and A the
return maps to the set A under f and , we see that, tautologically, fJ coincides
with the return map to J0 under fJ [K [L , and likewise J coincides with the
return map to J under J [K [L. (Every point of J^0 has to return to J^0 because J^0
is a lift of a cross section; and the same holds for K^ 0, and L^ 0 .) It suces then to
0
0
0
16
0
show that fJ [K [L induces on the union of three tori (J0= ) [ (K0 = ) [ (L0 = )
a map that is conjugated to J [K [L via the homeomorphism induced by h^ .
To see that, recall the diagram (3.4) from Section 3 to the eect that h^ con^
^
= ^T^^ ()
jugates f to f := T^ ^T^^ ()
^ T . Hence, fJ [K [L is conjugated by
^
^
^
h^ to fJ^ [K^ [L^ . However, ^T^^ ()
^ = Id except on Jw [ Ku [ Lv where it equals
:= ^JJ^^w[K[^K^[uL^[L^ v (cf. Figure 4.2). Therefore, on ^ n (J^w 1 [ K^ u 1 [ L^ v 1 ), f
coincides with T^ , and one easily computes by using (v) of Fact 3.4 that
8
^ w3
^
>
< T (p) if p 2 J0 ;
(4.1)
fJ^ [K^ [L^ (p) = > T^ u3(p) if p 2 K^ 0 ;
:
v
3
^
^
T (p) if p 2 L0 :
Upon descending to T3 , J^0, K^ 0 , and L^ 0 yield the cross sections J , K , and L; and
yields the return map J [K [L. The above formula shows then that fJ^ [K^ [L^
descends to the return map J [K [L. Thus fJ^ [K^ [L^ is conjugated to J [K [L, as
needed.
To nish the proof, we have to argue that if f is A-renormalizable then the
suspension ow of f has a triple of clean cross sections J , K , and L (in the appropriate cohomology classes) so that the map h (in the denition of renormalizability)
can be realized as the holonomy between ^ and S^ (as constructed in Section 3).
Let f := h f h 1 : ^ ! ^ . By denition of renormalizability, f = T^ on
^ n (J^w 1 [ K^ u 1 [ L^ v 1). Take g : J^w [ K^ u [ L^ v ! J^0 [ K^ 0 [ L^ 0 so that
fjJ^w [K^ u [L^ v = g T^ . By a routine construction there is a ow on Q0 such
that it enters Q0 through J^w [ K^ u [ L^ v , exits through J^0 [ K^ 0 [ L^ 0 , and realizes
g as the associated holonomy. After extending the ow equivariantly from the fundamental domain Q0 to all of R 3 =, we obtain nothing else than the cyclic covering
of the suspension ow of f. The quotient ow on T3 is then conjugated to the
suspension ow of f . By construction, the quotients J := J^0 =Z3, K := K^ 0 =Z3, and
L := L^ 0 =Z3 are the sought after clean cross sections. 2
Any map of the form RJ (f ) 1 : T2 ! T2 (where : J0= ! T2 is a
homeomorphism) is called a (u; v)-renormalization of f . Often it is more natural is
to consider the torus (J0 [ K0 [ L0 )= obtained by identifying the sides of the
hexagon J0 [ K0 [ L0 (c.f. Figure 4.1) and the map RJ [K [L : (J0 [ K0 [ L0 )= !
(J0 [ K0 [ L0 )= induced by the return map fJ [K [L to J0 [ K0 [ L0 (c.f.
(4.1)). Any rescaling of RJ [K [L to the original T2 is called an A-renormalization
of f . Figure 4.3 compares the renormalization on T2 with the renormalization on
T. (Here, pqnn and pqnn are two consecutive continued fraction convergents of the
rotation number , i.e. a Farey pair for .) The picture for d > 2 is readily obtained
from a generic projection of the d + 1-dimensional cube into R d .
The proof of Proposition 4.3, specically (4.1), relates A-renormalizations of f
to the holonomy between the entrance and exit faces of the fundamental domain
0
0
0
1
0
0
0
0
3
0
0
3
0
0
0
0
3
3
3
3
3
0
3
3
0
0
0
3
3
1
3
3
1
3
3
3
0
0
3
3
1
3
3
3
0
0
0
0
0
0
+1
+1
17
0
0
0
0
0
0
fv
f qn (0)
3
fu
3
f qn
fw
3
+1
0
(0)
f qn
f qn
+1
Figure 4.3: Renormalization in dimension 2 and 1.
Q0 and yields the following corollary.
Proposition 4.5 In the context of Proposition 4.3, any A-renormalization is conjugated to the Poincare return map to a cross section cohomologous to the ane
torus that is the quotient of the plane through u, v, and w.
We leave the proof as an exercise and remark only that a suitable cross section can
be readily built by cutting J , K , and L along their mutual intersections and by
gluying (and slightly deforming) the resulting pieces togeather.
To nish, let us comment that any concrete renormalization scheme would entail selecting representatives from the whole conjugacy class of A-renormalizations.
In that connection, we note that when f is conformal (a translation), then there
is a preferred conformal conjugacy class within all A-renormalizations:
Remark 4.6 If the torus T2 acted upon by an A-renormalizable map f 2 Di0(T2 )
is equipped with a conformal structure with respect to which f is conformal, then
there are natural induced conformal structures on the renormalized tori J0 = ,
K0 = , L0 = , and (J0 [ K0 [ L0 )= . These structures are independent of the
map h in Denition 4.1.
Sketch of Proof of Remark 4.6. By the measurable Riemann mapping theorem,
it suces to dene the conformal structure almost everywhere. The interiors of
J0, K0, and L0 carry the conformal structure induced from T2 by the inclusion
embedding. This induces a.e. conformal structure on J0= , K0 = , L0 = , and
(J0 [ K0 [ L0 )= . It is left to see that these conformal structures are independent
of the choice of h in Denition 4.1. Consider J0 and J0 arising from two dierent
such choices. By Proposition 4.3, RJ is conjugated to the return map J to the
cross-section J , and the conjugacy is induced by the holonomy from J0 to J^0. In
the suspension covering R 3 =, there is a unique ow-line through each interior
point of J0. Therefore, by considering all the ow-lines in R 3 = that avoid the
boundaries of J0 and J0, we get a natural a.e. dened mapping between J0 and J0.
This mapping is conformal. Similar arguments work for K0 and L0 . 2
0
18
5 Clean Triples of Cross Sections
Following the outline of the proof of Theorem 1.2 in Section 2, we turn to the
existence of clean triples of cross sections for ows on T3 and prove Theorem 2.2.
Below we continue to use the standard identication of H1(T3 ) and H 1(T3 ) with
R 3 of Cartesian coordinates x; y; z. Let us assume that the cohomology classes of
sections J , K , and L in the formulation of Theorem 2.2 are represented by the
1-forms dz, dx, and dy, respectively. (This can always be assured by conjugating
the ow by a linear automorphism of T3 .) We may also x on J , K , and L the
orientation induced by those cohomology classes and assume that the ow pierces
J , K , and L in the positive direction. This forces the rotation set to be in the
positive octant:
() f(x; y; z) : x; y; z > 0g;
(H)
where () is dened as the rotation set of the time-one map of the lifted ow
~ : R 3 R ! R 3 ; namely, () := (~1 ) (c.f. Section 2 in [15]).
In preparation for the proof of Theorem 2.2, note that, by virtue of Thom's
transversality theorem, J , K , and L can always be perturbed so that they are
mutually transversal. In such case, J \ K , K \ L, and L \ J are 1-dimensional
compact manifolds, i.e. nite unions of disjoint simple loops, which we call intersection loops. Likewise, the set J \ K \ L consists of nitely many points, which
we call triple points. Consider for a moment just one pair of cross sections,
P say L
and J , and let k 's be the connected components of L \ J so that L \ J = rk=1 k .
Observe that each k is a simple loop that is either null-homotopic or of homology
type (1; 0; 0), common to both J and L. In fact, if : H2(T3 ) H2 (T3 ) ! H1(T3 )
is the intersection product, we have
r
X
k=1
[k ]H (T ) = [L \ J ]H (T ) = [L]H (T ) [J ]H (T ) = (1; 0; 0):
1
3
1
3
2
3
2
3
(5.1)
Hence, among the essential (i.e. homotopically non-trivial) intersection loops k ,
all but one occur in homologically opposing pairs. Roughly speaking, those pairs
result from L and J getting folded through each other as depicted in Figure 5.2.
The null-homotopic k 's result, in turn, from L and J bumping into each other as
depicted in Figure 5.1. Elimination of such folds and bumps was carried out in [15]
to show existence of clean pairs of cross sections. Here we extend the argument to
handle the more complex intersection possibilities arising for three cross sections.
The strategy will be to inductively diminish the complexity of a triple (J; K; L)
dened as c(J; K; L) = (c0; c1) with
c0 := b0 (J \ K \ L) 1
(5.2)
c1 := b0 (J \ K ) + b0 (K \ L) + b0 (L \ J ) 3
19
(5.3)
where b0 stands for the number of connected components (the 0th-Betti number).
Clearly, (J; K; L) is clean if and only if c(J; K; L) = (0; 0).
Proof of Theorem 2.2: We assume that J , K , and L are transversal as in the
preceding discussion. We shall describe four modications, called reductions, each
yielding another transversal triple of cross sections with either c1 or c0 diminished.
Each reduction will be applicable under a dierent hypothesis, but they all will
isotope one of the cross sections inside a judiciously chosen topological 3-disk or
a solid torus inside T3 . The proof will culminate in combining the reductions to
lower the complexity to (0; 0).
The rst three reductions go back to [15] so we give here a somewhat less detailed exposition. A null-homotopic intersection loop is called minimal if it bounds
a disk in one of J , K , or L that contains no other null-homotopic intersection
loops.
Simple Bump Reduction. Hypothesis: There is a minimal null-homotopic
intersection loop with no triple points.
To x attention, we assume that the null-homotopic loop, call it , appears among
the components of L \ J and the disk DJ bounded by in J contains no other
null-homotopic intersection loops. Other cases are treated completely analogously.
Let DL be the disk bounded by in L. DL [ DJ forms a piecewise-smoothly
embedded 2-sphere in T3 and thus bounds an embedded 3-disk B in T3 . This last
assertion follows from a version of the classical Schonies Theorem found in the
appendix in [15].
Since the ow is transversal to L and J , it either enters or exits B across the
entire two-dimensional interiors DLo and DJo ; and it cannot enter (or exit) through
both sets because the hypothesis (H) on the rotation set precludes (forward or
backward) invariance of B . By reversing the ow if necessary, we may assume
then that the ow enters B via DLo and exits B via DJo .
J
J
L
J
L
new L
new L
Figure 5.1: A Bump Removal.
We are now ready to modify L in three steps as follows (see Figure 5.1).
Step 1. [Surgery] Replace DL with DJ . Note that this modication can be
eected by an isotopic deformation along the ow lines supported in B : a
20
point p of DL is connected inside B with a unique point of DJ by the ow
line through p. (In particular, the isotopy class of L in T3 is unaected.)
Step 2. [Detachment] Replace L with its image under the time--map for
some small > 0 to be specied later (in the proof of Claim 5.1). This is
meant to detach L from DJ (and thus restore transversality of L and J ).
Step 3. [Smoothing] Perturb L to a smooth torus while preserving transversality to the ow and the two other cross sections. What needs to be done
here is a rounding of the (edge) singularity along := ( ) generated by
Step 1. The perturbation must be C 0-small and C 1-small away from ; how
small is determined in the proof of Claim 5.1. We shall skip the details of this
tedious but routine step (best performed after dieomorphically straightening a small tubular neighborhood of onto a neighborhood of the depicted
in Figure 5.1.)
The goal of simple bump reduction is summarized in the following claim.
Claim 5.1 The modication following steps 1 through 3 leaves complexity c0 unchanged and diminishes complexity c1 ; specically, the number of the minimal nullhomotopic intersection loops is diminished by at least one.
Proof of Claim 5.1: First note that DJo \ K = ; because (by the minimality of
) all the loops in J \ K intersecting DJo would have to be homotopically non-trivial
and thus would have to hit = @DJ , contradicting the hypothesis on the absence
of triple points on . By transversality, some neighborhood V of the disk DJ is free
of K . Consequently, the modication of L eected by Steps 1 and 2 does not aect
K \ L or J \ K \ L as long as > 0 is small enough so that t(DJ ) is contained in
V for 0 < t < . And this is still true after Step 3 if the perturbation is suciently
small. In particular, c0 is indeed unaected by this reduction.
At the same time, (J )\J = ; for all small > 0. In particular, (DJ )\J = ;
so that, after Step 2, L no longer meets J in some neighborhood of B ; somewhat
imprecisely, we say that is removed form L \ J (perhaps together with some
other loops in DL [ J ). Again, this persists through Step 3 if the perturbation is
suciently small. As a result, c1 is diminished because, in the complement of a
small neighborhood of B , the modication of L amounts to a small C 1-perturbation
so that the number of connected components of L \ J (in that complement) is
preserved by virtue of transversality. 2
Complex Bump Reduction. Hypothesis: There is a minimal null-homotopic
intersection loop that contains triple points.
This reduction proceeds exactly as the just described Simple Bump Reduction and
we assume that , DJ , DL and B are as before with the only dierence that now
contains a triple point. Thus DJ intersects K , and J \ K \ L will be aected.
Claim 5.2 Complexity c0 is diminished.
21
Note that we make no claim about c1 (and c1 may in fact increase).
Proof of Claim 5.2: The reasoning is very similar to that showing Claim 5.1,
so we are brief. First, transversality assures that L will not pick up new points
in J \ K \ L outside some small neighborhood of B provided > 0 and the
perturbation in Step 3 are suciently small. Second, because (J ) \ J = ;, the
modied L no longer meets J in some neighborhood of B ; in particular, the points
of \ K are removed from J \ K \ L. Thus c0 is diminished. 2
Fold Reduction. Hypothesis: c1 > 0 and there are no null-homotopic intersection loops. All intersection loops are essential and one of the intersections L \ J ,
J \ K , and K \ L contains more than one loop. For specicity (and with no loss of
generality), we assume that this is L \ J and we denote the components of L \ J
by 1; : : : ; r . Viewed on the two-torus J (with H1(J ) identied with R 2 by taking
the x and y-directions as the basis), k 's are disjoint (1; 0)-loops, which cut J into
a number of annuli. The situation is analogous for L.
Lemma 5.3 There are components i and j of L \ J and closed annuli AJ in J
and AL in L with @AJ = @AL = i [ j such that AJ [ AL bounds a solid torus B
in T3 . Moreover, one can require that AoJ \ L = ;
Proof: The idea is to make sure that AJ [ AL is a 2-torus that deforms to
a loop in T3 . The plan is to select i and j so that J and L intersect along
i and j with opposite signs (in the sense of intersection homology). This is
easily done at the level of lifts as follows. Consider some (connected) lifts J~ and
L~ of J and L. Note that J~ = J~ + Z Z 0 and L~ = L~ + Z 0 Z, so if
p 2 L~ + (k1; k2; k3) \ J~ + (l1; l2; l3) = L~ + (0; k2; 0) \ J~ + (0; 0; l3) for some k; l 2 Z3,
then p (0; k2; l3) 2 L~ (0; 0; l3) \ J~ (0; k2; 0) = L~ \ J~. Thus the total preimage
of L \ J under the natural projection to T3 is
(L~ + Z3) \ (J~ + Z3) = (L~ \ J~) + 0 Z Z:
It follows that L~ \ J~ must have more than one connected component because L \ J
is not a single loop.
Consider then two distinct components ~i and ~j of L~ \ J~ and the strips (i.e.
topological [0; 1] R ) A~J and A~L they bound in J~ and L~ . We may take ~i and
~j adjacent in J~ in the sense that no other component of L~ \ J~ sits in A~oJ . Upon
descending to T3 , we get two components i and j of L \ J , and annuli AJ J
and AL L with @AJ = @AL = i [ j . Also, i and j are adjacent in J so
that AJ is a connected component of J n L, which guarantees AoJ \ L = ;. Now,
A~J [ A~L is a piecewise-smooth cylinder in R 3 that is equivariant under Z(0; 1; 0)
(by the construction). Therefore, AJ [ AL is a a piecewise-smooth 2-torus in T3
embedded so that the induced map H1(T2 ) ! H1(T3 ) has rank one. By a version
of Alexander Torus Theorem (Theorem 8.10 [15]), AJ [ AL bounds a solid torus B
in T3 . 2
22
The ow is transversal to AJ and AL, and (by reversing time if necessary) we
may assume that it enters B via AoL and exits via AoJ . This hinges on the fact that
B cannot contain a forward or backward semi-orbit because (0; 1; 0) is precluded
from the rotation set () by the hypothesis (H).
i
j
J
L
i
j
J
new L
L
J
new L
Figure 5.2: A Fold Removal.
We modify L in three steps analogous to those in the previous two reductions
(see Figure 5.2).
Step 1. Replace AL with AJ (via an isotopy in B along the ow lines).
Step 2. Replace L with its image under the time--map for small > 0.
Step 3. Smoothen L by a small perturbation.
Claim 5.4 The complexity c0 is diminished.
Proof of Claim 5.4. By transversality, for small > 0, L will pick up no new
points in J \ K \ L outside some small neighborhood of B where the modication
amounts to a small C 1-perturbation. At the same time, i and j are removed
from L \ J ; therefore, the points of (i [ j ) \ K are removed from J \ K \ L and
c0 is diminished. Here we used that (i [ j ) \ K is nonempty (in fact, contains at
least two points) because i and J \ K have a non-zero intersection number in J ,
and so do j and J \ K . 2
Finally we describe the most involved reduction, which has no analogue in [15]
so we proceed more carefully.
Triple Reduction. Hypothesis: c1 = 0 and c0 > 0.
Since c1 = 0, J \ K , K \ L, and L \ J are simple closed loops. Note that where
any two of these loops intersect all three must intersect, and the intersection points
form the set of triple points, T := J \ K \ L. As in the previous reductions, the
plan is to nd a suitable 3-disk in T3 on a neighborhood of which one cross section
will be modied to diminish #T = c0 + 1.
Viewed in the torus J , J \ K and L \ J are two simple loops of homology classes
(0; 1) and (1; 0) that intersect transversally along T . Because #T > 1, we claim
23
that there must exist two subarcs J \ K and L \ J such that [ bounds
a disk DJ in J (c.f. Figure 5.3). Moreover, by taking such disk to be minimal with
respect to inclusion, we may require that L [ J meets and J \ K meets only at
the common endpoints of and , which endpoints we further denote by t and t0 .
The above two facts are easily shown via the Jordan Theorem applied in the plane
covering J . (In particular, t and t0 can be found as the adjacent intersection points
between some connected lifts of J \ K and L \ J | c.f. the proof of Lemma 5.3.)
By similar considerations of the way intersects K \ L in K , one can see that
of the two subarcs of the simple loop K \ L that terminate at t and t0 , one | call
it | is such that [ bounds a disk DK in K (c.f. Figure 5.3). Finally, being
the boundary of the disk DJ [ DK in T3 , [ is a contractible loop and therefore
bounds some disk DL in L. The union DJ [ DK [ DL is a piecewise smooth sphere
in T3 , which we denote by .
There is a 3-disk B bounded by in T3 . Observe the following restrictions on
how the interior B o can meet J [ K [ L (which may happen since nothing prevents
from containing many triple points).
Fact 5.5 (i) DJo \ (L [ K ) = ;;
(ii) DLo \ K = ;;
(iii) DKo \ L = ;;
(iv) \ T = \ T .
Proof: (i) Suppose that DJo \ L 6= ;. This means that the curve L \ J crosses
the boundary of DJ : L \ J \ ( [ ) 6= ;. However, being simple, L \ J cannot cross
its own subarc , and L \ J \ o = ; by the construction of (where o stands for
the one-dimensional interior of ). Likewise, one can see that DJo \ K 6= ; would
contradict J \ K and J \ K \ o = ;.
(ii) As above, @DL = [ , and K \ L can enter DL neither through because
K \ L nor through o because that would put a triple point in o (contradicting
J \ K \ o = ;).
(iii) Otherwise, the curve K \ L would have to cross @DK = [ , which is
impossible for a similar reason as in (ii).
(iv) Parts (i), (ii), (iii), imply that \ T ( [ [ ) \ T . However, ( [ ) \ T =
ft; t0g because o and o contain no triple points by construction. The equality
(iv) follows. 2
We set out to analyze \ J , which may be quite complicated if o \ T 6= ;, as
illustrated in Figure 5.3. The goal is to nd another 2-sphere 1 B that bounds
a 3-disk B1 in T3 with a property that B1 o \ (J [ K [ L) = ;. Let D be the family
of all the connected components of B \ J with the exception of DJ | which is
the only component of B \ J entirely contained in . The transversality between
J , K , and L, assures that D 2 D is a compact surface with a piecewise smooth
boundary contained in .
24
FK
FK
FJ
DK
DJ
U
U0
in the back
FK
FL
FK
Figure 5.3: The 3-disk B sliced by lifts of J , the graph G , and the sphere 1 . (J and
K are depicted as ane tori. The dashed intersection lines reveal the locus of L.)
Fact 5.6 Each D 2 D is a closed topological 2-disk.
Proof: Let B~ R 3 be a connected lift of B . In B~ , which is a topological 3-disk,
~ D~ J , D~ K , and D~ L of the sphere and DJ ,
there are also contained unique lifts ,
DK , and DL, respectively.
The plane K~ that is the connected lift of K containing D~ K separates R 3 . By
(i) and (ii) of Fact 5.5, D~ Jo and D~ Lo do not intersect K~ + Z(1; 0; 0) and thus the
sphere ~ = @ B~ = D~ J [ D~ K [ D~ L intersects K~ + Z(1; 0; 0) solely along the disk
D~ K K~ . It follows that B~ is entirely contained on one side of K~ and that B~
avoids K~ + (m; 0; 0) for m 6= 0 (c.f. Figure 5.3).
J~ \ K~
C1
D~
J~ \ L~
Figure 5.4: A typical element of D in J~ (depicted as the horizontal plane).
Now, x D 2 D. We have to show that D is a topological disk. Let D~ B~ be
25
a connected lift of D and J~ be the connected lift of J containing D~ . By denition
of D, @D ( n DJ ) \ J (J \ K ) [ (L \ J ) so that @ D~ (J~ \ (K~ + Z(1; 0; 0))) [
((L~ + Z(0; 1; 0)) \ J~) where L~ is a connected lift of L. By our previous argument,
@ D~ (J~ \ K~ ) [ ((L~ + Z(0; 1; 0)) \ J~), and D~ is entirely contained on one side of
J~\ K~ in J~ | see Figure 5.4. Therefore, on the force of the Jordan Theorem, there
is no J~\ K~ inside the Jordan curve C constituting the boundary of the unbounded
component of J~ n D~ . Hence, should there be another component C1 of @ D~ , we
would have C1 (L~ + Z(0; 1; 0)) \ J~ (since C1 sits inside C ). This contradicts
L \ J being an essential loop. It follows that C is the sole component of @ D~ , which
makes D~ a disk. 2
Thus each D 2 D is a 2-disk in B that intersects = @B along its boundary
circle and thus separates B into two sets the closures of which are again 3-disks.
Any one of those two new 3-disks may be further separated in an analogous manner
by another member of D and so on. (The above assertions of course depend on the
Schonies Theorem for piecewise smooth embeddings; however, we leave o their
routine proofs.) We need some rudimentary understanding of the combinatorics of
the resulting partition of B into 3-disks.
To be more precise, let U be the family of the closures of the connected components of B n J . We shall say that U1 2 U is adjacent to U2 2 U along D 2 D
i D @B U1 \ @B U2 , where the subscript B indicates that the boundary is taken
relative to B . I view of the preceding remarks, D collects exactly the connected
components of the relative boundaries @B U for all U 2 U ; and U1 2 U is adjacent
to U2 2 U along some D 2 D i U1 \ U2 6= ;. Let G be the (undirected) abstract
graph with vertices U and edges D where an edge D 2 D is joining U1 and U2 i
U1 is adjacent to U2 along D (see Figure 5.3).
Fact 5.7 G is a tree.
Proof: We have to show that G has no non-trivial cycles. Consider a nonbacktracking path in G , i.e. a sequence of vertices U0 ; U1; : : : ; Um+1 and a sequence
of edges D1; D2 ; : : : ; Dm such that Di joins Ui to Ui+1 and Di 6= Di+1 for i =
1; : : : ; m. Recall that Di separates B into two topological 3-disks whose closures
we denote Wi and Wi+ so that Wi contains Ui and Wi+ contains Ui+1 . Any two
of the four 3-disks fWi; Wi+1g either are disjoint or are contained in one another.
Since Di+1 Wi++1 and Di+1 Ui+1 Wi+, we have Wi+ \ Wi++1 6= ;; and therefore
Wi++1 ( Wi+ or Wi++1 ) Wi+. The latter inclusion is however precluded by Ui+1 Wi+ and Ui+1 6 Wi++1. Thus Wi++1 ( Wi+ and we see that W1+ ) W2+ ) : : : ) Wm+.
It follows that D1 6= Dm , which shows that the path cannot be a cycle. 2
The graph G always has a vertex U0 2 U that contains DJ ; we refer to U0 as
the root of G . Unless G consists of a single vertex, it also has ends, i.e. vertices
other than U0 that belong to only one edge.
Fact 5.8 Suppose that U 2 U is an end of G and that FJ = U \ J , FK = U \ K ,
and FL = U \ L.
26
(i) U is a topological 3-disk, U o \ (J [ K [ L) = ;, @U = FJ [ FK [ FL, and FJ
is a 2-disk;
(ii) for each of the sets FJo, FKo , FLo , the ow either exists U across the whole set
or enters U across the whole set.
Note that FK and FL may be disconnected (as the FK in Figure 5.3).
Proof: (i) U being an end of G means that there is a single D 2 D such that
D = @B U and therefore FJ = U \ J = @B U = D by the denition of U . Thus U is
the 3-disk bounded by the two 2-sphere formed in the union FJ [ ( n DJ ).
Let us argue that U o \ (J [ K [ L) = ;. That U o \ J = ; follows from the
denition of U . If U o \ L 6= ;, then B o \ L 6= ;, and we could use @B = =
DJ [ DK [ DJ and transversality of J , K , and L to infer L \ (DJo [ DKo ) 6= ;, which
contradicts (i) and (iii) of Fact 5.5. Similarly, U o \K = ; because K \(DLo [DJo ) 6= ;
would contradict (i) and (ii) of Fact 5.5.
Finally, denition of U implies that @U J [ K [ L, and so @U = FJ [ FK [ FL
follows immediately from U o \ (J [ K [ L) = ;.
(ii) The ow either enters or exits B across the whole 2-disk DKo by transversality
and connectedness of DK . Because U B with FKo DKo , at the points of FKo , the
ow enters (exits) U i it enters (exits) B , which implies the assertion (ii) regarding
FK . The arguments for FL is analogous. The case of FJ follows by transversality
and connectedness2 of FJ = D (supplied by (i)). 2
How the ow intersects U is further constrained by Wa_zewski's theorem (see
e.g. [3].) Let U + be the exit set of U , i.e., U + := fp 2 U : 8>090<t< t(p) 62 U g.
Also, let U be the entrance set of U , i.e., the exit set under the reversed ow.
Fact 5.9 (i) U + and U are closed subsets of U ;
(ii) U is a Wa_zewski set under the ow and its inverse3;
(iii) U and U + are topological 2-disks, each coinciding with one set or a union
of two sets from the collection fFJ ; FK ; FL g.
Proof: (i) We shall argue only for U + as the same arguments can be applied to U after reversing the ow. It suces to show that W + := fp 2 @U :
9>080<t< t (p) 2 U g is open in @U . That W + \ (FJo [ FKo [ FLo ) is open follows
from transversality (of the ow to J , K , and L), which leaves the openness at the
points of the singular locus: (FJ \ FK ) [ (FK \ FL ) [ (FL \ FJ ).
We consider rst a (triple) point p 2 FJ \ FK \ FL . By routine analytics, there is
a small neighborhood V of p on which J \ V , K \ V , and L \ V can be straightened
via a dieomorphism of V into subsets of the three standard coordinate planes in
2 This is the only place where it is critical that U is an endpoint, not some other element of U
| all of which are topological 3-disks.
3 That is (
+) constitutes an index pair in the sense of Conley
U; U
27
R 3 . In particular, the complement of J [ K [ L in V has eight components, each
occupying (mapping to) a dierent octant in R 3 . Observe that U \ V occupies
only one of the octants because U o is free of J [ K [ L by (i) of Fact 5.8. By
transversality of the ow to J , K , and L, if p 2 W +, then the vector generating
the ow at p must point strictly inside the octant occupied by U , and it follows
that W + is open at p.
Now consider p 2 (FJ \ FK ) [ (FK \ FL) [ (FL \ FJ ) n (FJ \ FK \ FL). Again,
two of the sets J , K , and L meet at p, and there is a neighborhood V of p on
which these two sets straighten out into subsets of two of the standard coordinate
planes in R 3 . Arguing as before | this time in terms of the quadrants associated
with the two planes | we see that U \ V must occupy only one of the quadrants
and that, if p 2 W +, then W + is open at p.
(ii) This amounts to realizing that, since U and U are closed, the axioms a) and
b) dening Wa_zewski sets in denition 2.2 on page 24 in [3] are automatically satised.
(iii) As a consequence of the hypothesis (H) on the rotation set, the maximal invariant subset of U is empty, be that under the ow or its inverse. Wa_zewski's
theorem, applicable on the force of (ii), asserts that both U + and U are deformation retracts of U . Hence, U + and U are both contractible, which makes them
2-disks. The assertion (iii) follows by taking into account that each of the sets FJo,
FKo , and FLo is contained in either U + or U by (iii) of Fact 5.8. 2
We are now ready to carry out the reduction. To dene a suitable 3-disk B1
(for localizing the isotopy), we consider the tree G . Either G has only one vertex
B , in which case we set B1 := B , or there is an end U of G as in Fact 5.9, in which
case we set B1 := U . The boundary of B1 is made of three sets FJ = B1 \ J ,
FK = B1 \ K , and FL = B1 \ L. (If B1 = B all three sets are 2-disks, whereas
if B1 = U only FJ must be a 2-disk and FK and FL may be disconnected.) In
any case, taking into account Fact 5.9, the ow enters and then exits B1 across
two 2-disks, each coinciding with one set or a union of two sets from the collection
fFJ ; FK ; FLg. For specicity, after perhaps reversing the ow, we shall assume
that enters B1 via FLo and exists via (FJ [ FK )o . The other cases are handled
completely analogously by simply permuting the notation pertaing to J , K , and
L. Note that @FL cannot be entirely contained in any one of the essential loops
L \ J or K \ L and therefore @FL contains some (at least two) triple points. We
shall modify L so that those triple points are removed via the familiar three steps:
Step 1. Replace FL with FJ [ FK via an isotopy in B1 (along the ow lines).
Step 2. Replace L with its image under the time--map for small > 0.
Step 3. Smoothen L.
Claim 5.10 The modication following the steps 1 through 3 reduces c0.
28
The proof of the claim is virtually the same as that of Claim 5.2, we skip it.
Conclusion of the proof of Theorem 2.2: We nish the proof by combining the
four reductions described above to yield a triple of cross sections with c0 = c1 = 0
(i.e. a clean triple). By construction, the reductions preserve the isotopy classes
of the cross sections (as well as their mutual transversality).
First consider the special case when already c0 = 0. Then each of J \ K , K \ L,
or L \ J must contain exactly one essential loop by considerations of intersection
homology (c.f. (5.1)). If also c1 = 0 we are done. Otherwise, c1 > 0 and there are
null-homotopic intersection loops, all free of triple points because c0 = 0. After
picking a minimal such loop, simple bump reduction lowers c1 while preserving
c0 = 0 (see Claim 5.1). Iteration of this process leads to c0 = c1 = 0.
We now turn attention to the case when c0 > 0. In view of the preceding
discussion, we will be done by induction if we succeed in lowering c0 (while perhaps
increasing c1). If c1 = 0, then c0 is lowered by triple reduction as asserted by
Claim 5.10. If c1 > 0, then simple bump reduction can be used to iteratively
remove the null-homotopic loops that are free of triple points without aecting c0
(see Claim 5.1). This may yield c1 = 0, a case we already resolved, or we may still
have c1 > 0. If c1 > 0, then either there is a minimal null-homotopic intersection
loop with triple points and we can apply complex bump reduction to reduce c0 (see
Claim 5.2), or there are no null-homotopic intersection loops and we are in position
to lower c0 via fold reduction (see Claim 5.4). 2
6 Appendix: Auxiliary Results
In Section 3, we use the following classical connection between ux and volume
(which, for lack of a convenient literature reference4 , we supply with a proof).
Theorem 6.1 (Reynolds' Transport Theorem) Let M be a Riemannian
manifold with the volume form and let : N ! M be a smooth submanifold
of codimension 1. Suppose that : R M ! M is a ow of a vector eld X on
M and N t := t (N ). If
:=
t[
=t0
N t ftg M R
t=0
and : M R ! M is the natural projection, then
Z
( ) =
Z t0
0
Flux of X through N t dt:
4 c.f. page 11 in Fundamental Mechanics of Fluids by I.G. Currie
29
(6.1)
If j : ! M is an embedding, then the left hand side of the formula (6.1)
can be interpreted as the signed volume (trapped between N 0 and N t ), which is
how we use the result in Section 3.
Proof. Let t := t and g : N R ! M R be given by (y; t) 7! (t(y); t).
We have
Z
Z
( ) =
g ( ):
0
N [0;t0 ]
The ux through N t equals5
Z
Nt
iX () =
Z
N
(t)(iX ())
where iX is the inner multiplication by X . It suces then to show the following
equality of n-forms on N R :
g () = (t )(iX ()) ^ dt:
(6.2)
At a point (y; t) 2 N R, a tangent vector to N R has the form v + a @t@ where
v is tangent to N . The image of v + a @t@ under the dierential of g is
@ = Dtv + aX
D( g)(y; t) v + a @t
where the spatial derivative Dt is evaluated at (y; t) and X is taken at t(y). Thus
we verify (6.2) by evaluating the forms on n-tuples of vectors:
n @
= Dtvi + aiX ni=1
g () vi + ai @t
i=1
n n
X
@
t
t
t
= D v1 ; : : : ; X ; : : : ; D vn aj = (( ) (iX ()) ^ dt) vi + ai @t
:
i
=1
j =1
2
In Section 2, we used the following lemma from dierential topology.
Lemma 6.2 If (J; K; L) is a clean triple of C 1 -smoothly embedded tori in T3 , then
there exist a homotopic to the identity C 1-dieomorphism h : T3 ! T3 that maps
each of J , K , and L into an ane two torus in T3 .
This result can be derived from the classical Schonies Theorem as suggested below.
Sketch of Proof: From the denition of a clean triple there is a basis u; v; w 2 Z3
of Z3 over Z such that J , K , and L are cohomologous to the ane tori obtained
5 See page 411 in Manifolds, Tensor Analysis, and Applications by Abraham, Marsden, Ratiu.
30
as the quotients of the planes lin(u; v), lin(v; w), and lin(w; u), respectively. With
no loss of generality we assume that u = (1; 0; 0), v = (0; 1; 0), and w = (0; 0; 1).
Let j; k; l : T2 ! T3 be embeddings such that j (T2 ) = J , k(T2 ) = K , and
l(T2 ) = L. The preimages j 1 (J \ K ) and j 1(L \ J ) that are two transversal
essential loops in T2 that intersect transversally at a single point. By perhaps
precomposing j with an appropriate dieomorphism of T2 , we may secure that
j 1(L \ J ) and j 1(J \ K ) are the ane loops covered by lin((1; 0)) and lin((0; 1)),
respectively. (This amounts to an analogue of the lemma under consideration in
one lower dimension, with a similar but simpler proof than what follows.) Likewise,
we shall require that k 1(J \ K ) and k 1(K \ L) are covered by lin((1; 0)) and
lin((0; 1)) and l 1(K \ L) and l 1(L \ J ) are covered by lin((1; 0)) and lin((0; 1)),
respectively.
Let Q = [0; 1]3 and ~j ; k~; ~l : R 2 ! R 3 be the lifts of j; k; l normalized so that
j (0) = k(0) = l(0) = 0. Consider the map g : @Q ! R 3 dened by the following
equalities where x; y 2 [0; 1]
g(x; y; 1) (0; 0; 1) = g(x; y; 0) = ~j (x; y)
(6.3)
g(1; x; y) (1; 0; 0) = g(0; x; y) = k~(x; y)
(6.4)
g(x; 1; y) (0; 1; 0) = g(x; 0; y) = ~l(y; x):
(6.5)
Roughly speaking, g is obtained by pasting together two copies of each j , k, and
l. The fact that (J; K; L) is a clean triple assures that g is a piecewise smooth
embedding of the piecewise smooth sphere @Q into R 3 . Therefore, g is not wild,
say it extends to a bicollared neighborhood of @Q, and the Schonies Theorem
(see [2]) assures that g extends to an embedding G : Q ! R 3 . Furthermore, G
extends to a Z3-equivariant map F : R 3 ! R 3 by setting F (p) := G(p + v) for all
p 2 R 3 where v 2 Z3 is taken such that p + v 2 Q. Because G(Q) is a fundamental
domain for the action of Z3, F is a homeomorphism. In fact, little extra care
in extending g guarantees that F is a dieomorphisms. The Z3-quotient of F ,
f : T3 ! T3 is a homotopic to the identity torus dieomorphism. The the sought
after dieomorphisms is h := f 1. 2
References
[1] D.V. Anosov and A.B. Katok. New examples of ergodic dieomorphisms of smooth
manifolds. Trans. Moscow Math. Soc., 23:1{36, 1970. (translated from Russian).
[2] G. E. Bredon. Topology and Geometry. Springer, New York, 1993.
[3] C. Conley. Isolated invariant sets and the Morse index, volume 38 of CBMS Regional
Conference Series in Mathematics. American Mathematical Society, Providence,
R.I., 1978.
[4] W. de Melo and S. van Strien. One-dimensional dynamics, volume 25 of Ergebnisse
der Mathematik und ihrer Grenzgebiete (3). Springer, Heidelberg, 1993.
31
[5] P. Fogg. Substitutions in Dynamics, Arithmetics and Combinatorics. Lecture Notes
in Mathematics. Springer-Verlag, 2002. to appear.
[6] D. Fried. The geometry of cross sections to ows. Topology, 21:353{371, 1982.
[7] H. Furstenberg. Strict ergodicity and transformations of the torus. Amer. J. of
Math., 83:573{601, 1961.
[8] M. R. Herman. Sur la conjugaison dierentiable des dieomorphismes du cercle a
des rotations. Inst. Hautes Etudes
Sci. Publ. Math., 49:5{233, 1979.
[9] M. R. Herman. Une methode pour minorer les exposants de lyapunov et quelques
exemples montrant le caractere local d'unn theoreme d'arnold et de moser sur le
tore de dimension 2. Comment. Math. Helv., 58:453{502, 1983.
[10] M. R. Herman. Some open problems in dynamical systems. Doc.Math., Extra
Volume ICM 1998, II:797{808, 1998.

[11] S. Kim and S. Ostlund.
Renormalization of Mappings of the Two-Torus. Phys. Rev.
Lett., 55 no. 11:1165{1168, 1985.
[12] H. Koch. A renormalization group for Hamiltonians, with applications to KAM tori.
Erg. Th. & Dyn. Sys., 19 no. 2:475{521, 1999.
[13] H. Koch. On the renormalization of hamiltonian ows, and critical invariant tori.
mp arc preprint 01-191, 2001.
[14] D. V. Kosygin. Multidimensional KAM theory from the renormalization group
viewpoint. Adv. Soviet Math., 3:99{129, 1991. (Translated from the Russian).
[15] J. Kwapisz. A priori degeneracy of one-dimensional rotation sets for periodic point
free torus maps. preprint, to be published in TAMS, 2000.
[16] J. Kwapisz. Renormalizing torus and annulus maps. in progress, 2002.
[17] M. Misiurewicz and K. Ziemian. Rotation sets for maps of tori. J. London Math.
Soc., 40:490{506, 1989.
[18] A. Norton and D. Sullivan. Wandering domains and invariant conformal structures
for mappings of the 2-torus. Ann. Acad. Sci. Fenn. Math., 21:51{68, 1996.
[19] A. Norton and J. Velling. Conformal irregularity for denjoy dieomorphisms of the
2-torus. Rocky Mountain J. Math., 24:655{671, 1994.
[20] N. Tongring V. Baladi, D. Rockmore and C. Tresser. Renormalization on the ndimensional torus. Nonlinearity, 5:1111{1136, 1992.
[21] R. F. Williams. The Penrose, Amman, and DA Tiling Spaces are Cantor Set Fiber
Bundles. Erg. Th. & Dyn. Sys., 21 no. 6:1883{1901, 2001.
32