Instituto de Matemática e Estatística Curso de Doutorado em Matemática
Coordenação de Pós Graduação em Matemática
FERNANDA PEREIRA RODRIGUES
ROTATIONAL DEVIATIONS FOR
MINIMAL TORUS HOMEOMORPHISMS
Orientador: Alejandro Kocsard
NITERÓI FEVEREIRO/2016
FERNANDA PEREIRA RODRIGUES
ROTATIONAL DEVIATIONS FOR MINIMAL TORUS HOMEOMORPHISMS
Tese apresentada por Fernanda Pereira
Rodrigues ao Curso de Doutorado em
Matemática - Universidade Federal Fluminense, como requisito parcial para a obtenção do Grau de Doutor. Linha de Pesquisa: Sistemas Dinâmicos
Orientador: Alejandro Kocsard
Niterói 2016
Ficha catalográfica elaborada pela Biblioteca de Pós-graduação em Matemática da UFF
R696 Rodrigues, Fernanda Pereira
Rotational deviations for minimal torus homeomorphisms / Fernanda Pereira Rodrigues. – Niterói, RJ : [s.n.], 2016.
50 f.
Orientador: Prof. Dr. Alejandro Kocsard
Tese (Doutorado em Matemática) – Universidade Federal Fluminense, 2016.
1.Homeomorfismos do toro. 2.Homeomorfismos minimais. 3.Funções no toro. I. Título.
FERNANDA PEREIRA RODRIGUES
ROTATIONAL DEVIATIONS FOR MINIMAL TORUS HOMEOMORPHISMS
Tese apresentada por Fernanda
Pereira Rodrigues ao Curso de
Doutorado em Matemática - da Universidade Federal Fluminense, como requisito parcial para a obtenção do Grau de Doutor. Linha de Pesquisa: Sistemas Dinâmicos.
Aprovada em: 25/02/2016
Banca Examinadora
_______________________________________________ Prof. Alejandro Kocsard - Orientador
Doutor – Universidade Federal Fluminense
_______________________________________________ Prof. Andrés Koropecki - Membro
Doutor – Universidade Federal Fluminense
_______________________________________________ Prof. Fábio Armando Tal - Membro
Doutor – Universidade de São Paulo
___________________________________________ Prof. Rafael Potrie Altieri - Membro
Doutor – Universidad de la República, Uruguai
_______________________________________________ Prof. Salvador Addas-Zanata - Membro
Doutor – Universidade de São Paulo
_______________________________________________ Prof. Sebastião Marcos Antunes Firmo - Membro
Doutor – Universidade Federal Fluminense
Contents
Agradecimentos . . . 1
Resumo . . . 2
Abstract . . . 3
1 Introduction 4 2 Notations and preliminaries 10 2.1 Maps, topological spaces and groups . . . 10
2.2 Euclidean spaces and tori . . . 11
2.3 Minimality . . . 11
2.4 Surface topology . . . 12
2.5 Groups of homeomorphisms . . . 12
2.6 Rotation set and rotation vectors . . . 13
2.7 Directional rotational deviations . . . 14
2.8 Theorem of Gottschalk and Hedlund . . . 15
2.9 Theorem of Oxtoby and Ulam . . . 15
2.10 Hamiltonian homeomorphisms . . . 16 3 Minimal Homeomorphisms 17 3.1 A general result . . . 17 3.2 Bounded deviations . . . 19 3.2.1 Pseudo-foliation . . . 19 3.2.2 Semiconjugacy . . . 27 4 Unbounded deviations 29 4.1 The fiberwise Hamiltonian skew-product and the stable set at infinity . . . 29
4.1.1 Motivation . . . 29
4.1.2 Definition and properties . . . 30
4.2 The minimal case . . . 38
4.2.1 Main result . . . 45
Agradecimentos
Agradeço à minha família por todo o apoio, carinho e incentivo. Obrigada por estarem sempre ao meu lado, cuidando de mim nos momentos em que precisei e comemorando comigo cada objetivo alcançado.
Ao Professor Alejandro Kocsard, agradeço pela orientação no doutorado, por todas as idéias e conversas sobre este trabalho. Aprendi muito em cada uma delas.
Agradeço à Professora Isabel Lugão Rios por ter me orientado durante a graduação e mestrado e pelo incentivo para que eu desse continuidade aos meus estudos.
Ao Professor Andres Koropecki, agradeço pelas observações e importantes sugestões sobre este trabalho.
Gostaria de agradecer também aos professores do IME da UFF. Aos que me deram aulas, agradeço por todos os ensinamentos. Aos que não tive a oportunidade de ser aluna, mas conheci ao longo destes anos, agradeço por terem sido sempre tão atenciosos comigo.
Aos colegas e amigos que convivi durante todo este tempo na UFF, agradeço à Marcele, Flávia e Gladys por terem compartilhado comigo tantos momentos durante minha graduação e, entre os amigos que acompanharam a minha trajetória na pós-graduação, agradeço especialmente à Sonia, Romulo, Vítor, André, Laurent, Daniel, Claudia, Mauro, Jaime, Priscilla, Jaqueline e Cristhabel.
Resumo
Seja f um homeomorfismo do toro homotópico à identidade e ˜f : R2 → R2
um levantamento de f . Generalizando o número de rotação introduzido por Poincaré para aplicações do círculo, Misiurewicz e Ziemian em [MZ89] definiram um conjunto compacto e convexo ρ( ˜f ) ⊂ R2, denominado conjunto
de rotação, que reflete muitas propriedades dinâmicas de f .
Este conjunto tem interior vazio se f não possui pontos periódicos e, portanto, podemos encontrar uma reta contendo o conjunto de rotação.
Nesta tese, consideraremos o caso de homeomorfismo minimal, i.e. um homeomorfismo com todas as órbitas densas no toro. Se v ∈ R2 é um vetor
perpendicular a reta contendo ρ( ˜f ), então a projeção ortogonal de ρ( ˜f ) na
direção de v é um único ponto Dρ( ˜f ), vE.
Dizemos que f tem desvio rotacional limitado na direção de v, quando
D
˜
fn(z) − z − nρ( ˜f ), vE
é uniformemente limitado para todo z ∈ R2 e n ∈ Z.
O objetivo do nosso trabalho é estudar a dinâmica de f considerando desvios rotacionais limitados ou ilimitados. Provamos que a existência de uma partição do toro invariante por f , formada por conjuntos conexos e essenciais com interior vazio, é uma condição necessária e suficiente para desvios limitados.
Além disso, mostramos que, se o conjunto de rotação é um segmento com inclinação racional, então temos a seguinte dicotomia: ou f possui desvio limitado ou todo conjunto aberto se espalha em todas as direções.
Abstract
Let f ∈ Homeo0(T2) be a 2-torus homeomorphism which is homotopic to
the identity and ˜f ∈ Homeo(R2) be a lift of f . Generalizing the well-known Poincaré rotation number for circle homeomorphisms, Misiurewicz and Ziemian defined in [MZ89] the so called rotation set ρ( ˜f ) ⊂ R2, which is
always compact and convex and reflects many dynamical properties of f . When f has no periodic points, the rotation set ρ( ˜f ) has empty interior,
and hence, one can find a straight line containing the rotation set.
In this thesis we consider the case where f is a minimal homeomorphism, i.e. every f -orbit is dense in T2. If v ∈ R2 denotes a vector perpendicular to
the line containing ρ( ˜f ), so the orthogonal projection of ρ( ˜f ) on the direction
of v is just a point Dρ( ˜f ), vE.
We say that f has bounded rotational deviation in the direction of v, when
D
˜
fn(z) − z − nρ( ˜f ), vE
is uniformly bounded for every z ∈ R2 and n ∈ Z.
Our main goal is to study the dynamics of f considering the boundedness or unboundedness of rotational deviations. We show that the existence of an invariant partition of T2, formed by connected unbounded sets with empty
interior, is a necessary and sufficient condition for bounded deviation. Moreover, we prove that, if the rotation set is a segment with rational slope, we have the following dichotomy: either f has bounded deviation, or every open set mixes in all directions.
Chapter 1
Introduction
If f : T → T is an orientation preserving homeomorphism of the circle, the most important dynamical invariant of f is the well-known rotation number of Poincaré: considering a lift ˜f : R → R of f , the rotation number of ˜f is
defined as ρ( ˜f ) = lim n→∞ ˜ fn(x) − x n .
He proved that the limit exists and is independent of the point x ∈ R. The rotation number has an important relation with the dynamics of f : ρ( ˜f ) is
rational if and only if f exhibits a periodic orbit; and if ρ( ˜f ) is irrational, f
is semiconjugate to the irrational rotation by ρ( ˜f ).
Moreover, in the one-dimensional case, it is well-known that f has bounded rotational deviations, i.e., | ˜fn(x) − x − nρ( ˜f )| is uniformly bounded for every
x ∈ R and n ∈ Z.
Misiurewicz and Ziemian [MZ89] introduced a generalization of the ro-tation number of Poincaré for homeomorphisms of Tm = Rm/Zm, with any
m ≥ 1, which are homotopic to the identity: given any homeomorphism f : Tm → Tm homotopic to the identity, consider a lift ˜
f : Rm → Rm of f .
Then, the rotation set ρ( ˜f ) is the set of the limits of all convergent sequences
˜ fni(x i) − xi ni ! i∈N , where xi ∈ Rm and ni → ∞.
Unlike the one-dimensional case, in general this set does not reduce to a point, but it is always non-empty, compact and connected. When m = 2, then ρ( ˜f ) is a convex subset of the plane. In this work, we just consider the
two-dimensional case. So, either ρ( ˜f ) has interior points, or it is contained
in a straight line of R2. When ρ( ˜f ) reduces to a single point, f is said to be
Now, consider a pseudo-rotation f : T2 → T2 and let ρ( ˜f ) = {ρ} be its
rotation set. In this case, we say that f has bounded rotational deviations if there is a constant C > 0 such that
˜ fn(z) − z − nρ ≤ C,
for all n ∈ Z and z ∈ R2. Unlike the one-dimensional case, the property of
bounded rotational deviations does not hold in general on higher dimensions. When f is an area-preserving pseudo-rotation of T2 with bounded
ro-tational deviations, Jäger [J¨09] proved a classification similar to the one-dimensional case:
1. If ρ is totally irrational, then f is semi-conjugate to Rρ;
2. If ρ is neither totally irrational nor rational, then f has a periodic circloid;
3. ρ is rational if and only if f has a periodic point.
The concept of circloid was introduced in [J¨09]: a subset C ⊂ T2 which is
compact and connected, essential (not contained in any embedded topological disk), with a connected complement which contains an essential simple closed curve and does not contain any strictly smaller subset with these properties. The third item of Jäger’s result was proved previously by Franks in [Fra95].
There are area-preserving pseudo-rotations with unbounded rotational deviations. The Furstenberg’s construction of a minimal non-uniquely er-godic homeomorphism is an example with this property (see Section 4 of [JS06]). The Furstenberg’s example is given by a skew-product of a transla-tion on the circle, i.e. it is a map of the form
(x, y) 7→ (x + ω, y + h(x)),
where (x, y) ∈ T2 and h : T → T. Then notice that, in spite of it has unbounded deviations, this example has bounded deviations on the horizontal direction.
In [KK09], they show there are examples of pseudo-rotations such that each open set mixes in every homological direction. This property is called
weak spreading and it implies unbounded deviations in every direction.
It is interesting observe that, even the rotation set is a unique vector with rational coordinates, the map may have unbounded deviations. For example,
Koropecki and Tal [KT14a] constructed an area preserving diffeomorphism, which is irrotational (i.e. a pseudo-rotation with ρ( ˜f ) = {(0, 0)}) such that
the deviations are not bounded in any direction.
If the rotation set is a non-degenerate segment, Franks and Misiurewicz [FM90] conjectured that ρ( ˜f ) is one of the following:
(i) A line segment with rational slope passing through some point of Q2;
(ii) A line segment with irrational slope having one end at point of Q2. Le Calvez and Tal [CT15] proved that a rotation set could not be a line segment with irrational slope and a point of Q2 in its interior. Artur Ávila
has announced a counter-example for the second case of conjecture having as a rotation set a segment with irrational slope such that ρ( ˜f ) ∩ Q2 = ∅.
We do not know if the first case of conjecture is true. In this work, we are particularly interested in this special case: ρ( ˜f ) with rational slope such
that ρ( ˜f ) ∩ Q2 = ∅.
If f is a periodic point free homeomorphism (i.e., Per(f ) = ∅), then Handel [Han89] proved that ρ( ˜f ) has empty interior. In this case, we know
that there exist v ∈ S1 and α ∈ R such that
ρ( ˜f ) ⊂ αv + Rv⊥.
We say that a point z0 ∈ T2 exhibits bounded v-directional rotational
deviations (bounded v-deviations for short) when there exists a real constant M = M (z0, f ) > 0 such that D ˜ fn(z0) − z0, v E − nα ≤ M, ∀n ∈ Z. (1.1)
Moreover, we say that f exhibits uniformly bounded v-directional
rota-tional deviation (uniformly bounded v-deviations for short) when there exists M = M (f ) > 0 such that
D
˜
fn(z) − z, vE− nα ≤ M, ∀z ∈ T2
, ∀n ∈ Z. (1.2) It is unknown whether every homeomorphism whose rotation set is a non-degenerate segment exhibits uniformly bounded rotational deviation in the perpendicular direction to the supporting line. However, when f is area preserving and such that ρ( ˜f ) is a segment of rational slope, parallel to v⊥
containing some point of rational coordinates, Guelman, Koropecki and Tal [GKT14] proved that f has uniformly bounded v-deviation.
Dávalos generalized this result and he proved that it remains true with-out the hypothesis of area preserving. In fact, he proved a stronger result, which give us the relationship between unbounded (1, 0)-deviation and pos-itive linear speed for some point. More precisely, he proved the following result:
Theorem ([Dáv13]). Let f : T2 → T2 be a homeomorphism homotopic to
the identity. Suppose that, for some lift ˜f : R2 → R2, the rotation set ρ( ˜f ) contains (0, a) and (0, b), with a < 0 < b. Then one of the following holds:
1. f has uniformly bounded (1, 0)-deviation;
2. there are x ∈ R2 and N > 0 such that pr1( ˜fnN(x) − x) > n for every
n ∈ N.
This result implies the generalization of the theorem of Guelman, Ko-ropecki and Tal. Indeed, if ρ( ˜f ) is a segment of rational slope, parallel to v⊥, containing some point of rational coordinates, there is a map A ∈ GL(2, Z) such that Aρ( ˜f ) is contained in a vertical line (see [KK08]), and we can
as-sume that ρ( ˜f ) is a vertical segment containing the origin. Since we have no
points with positive linear speed, then f has uniformly bounded deviation.
Moreover, in [Dáv13], it is proved that, if the rotation set ρ( ˜f ) is a
poly-gon whose extremal points are vectors with rational coordinates, then the distance d( ˜fn(x) − x, nρ( ˜f )) is uniformly bounded. This result is also
ob-tained as a consequence of Dávalos Theorem stated above.
A more general version of this result was proved by Le Calvez and Tal [CT15]: if ρ( ˜f ) has a non empty interior, then there is M > 0 such that
d( ˜fn(x) − x, nρ( ˜f )) < M,
for every x ∈ R2 and every n ∈ N.
In this work, we study the case that f is minimal. Notice that it is a particular case of periodic point free homeomorphism and then we have that
ρ( ˜f ) is a unique point or a segment. Moreover, from [Fra95], ρ( ˜f ) ∩ Q2 = ∅.
Although we will work with the minimal case, some techniques of our proofs are also valid for the more general case of periodic point free homeo-morphisms. This fact will be mentioned in Chapter 4.
First, we study the case that f has rotation set ρ( ˜f ) ⊂ αv + Rv⊥, v ∈ R2, and exhibits bounded v-deviation.
We define a pseudo-foliation of T2 as a partition of T2 such that each set of this partition has empty interior and its lift in R2 separates the plane in
exactly two connected components, which are both unbounded.
Thus, we use Gottschalk-Hedlund Theorem to show that, if f is minimal, the following properties are equivalent:
1. f has uniformly bounded v-deviation;
2. f has an invariant pseudo-foliation of T2 such that the lift of each
pseudo-leaf is contained in a strip with the direction of v⊥.
In this case, we say that the invariant pseudo-foliation has the asymptotic
direction of v⊥.
If v has rational slope, we obtain that f is semiconjugate to the rotation
Rα on T as a direct consequence of the construction of the invariant
pseudo-foliation.
Notice that the fibres of the semiconjugacy in general could be rather wild. In fact, Béguin, Crovisier and Jäger [BCJ15] constructed a minimal smooth pseudo-rotation which is semiconjugate to an irrational rotation, and the fibres are pseudo-circles.
Furthermore, we study the case that the rotation set is a nontrivial seg-ment with rational slope. Then there is a map A ∈ GL(2, Z) such that
Aρ( ˜f ) is contained in a vertical line, and we have that ρ(A ˜f A−1) is a point or vertical segment with ρ( ˜f ) ∩ Q2 = ∅. Therefore, it is sufficient to consider
ρ( ˜f ) ⊂ {α} × R, with α ∈ R \ Q.
We will prove that f is spreading if f has not bounded (1, 0)-deviation. The definition of spreading can be found in [KK09]:
Definition 1. A homeomorphism f : T2 → T2 is said to be spreading if for
a lift ˜f : R2 → R2 of f the following holds: for each open set U ⊂ R2, > 0
and N > 0, there is n0 ∈ N such that ˜fn(U ) is -dense in a ball of radius N
whenever n ≥ n0.
This property implies that f is topologically mixing. Our result can be stated as follows:
Theorem 1.0.1. Let f be a minimal homeomorphism homotopic to the
iden-tity with ρ( ˜f ) = {α} × [a, b], with α ∈ R \ Q and a < 0 < b. Then, either f has bounded (1, 0)-deviation, or f is spreading.
Thus, from the results above, we conclude that, if f a minimal homeomor-phism of T2 homotopic to the identity with rotation set ρ( ˜f ) = {α} × [a, b],
then we have the following classification:
1. If f has bounded (1, 0)-deviation, f is semiconjugate to an irrational rotation on T.
2. If f has unbounded (1, 0)-deviation, f is spreading.
Notice that, if there is a minimal homeomorphism f of T2 homotopic to
the identity such that ρ( ˜f ) is a segment with rational slope, then it will be
a counter-example for the first case of Franks-Misiurewicz conjecture stated above. Our result implies that, if there is a minimal counter-example f for this case of conjecture, and f has unbounded deviation in the perpendicular direction of ρ( ˜f ), then f will be minimal and topologically mixing.
This work is organized as follows. In Chapter 2, we will introduce some notations and preliminary results. In Chapter 3, we will consider minimal homeomorphisms of T2 with bounded v-deviations for every v ∈ R2.
More-over, in this chapter, we prove a general result about minimal homeomor-phisms, which will be an important tool in our proofs. In Chapter 4, we will introduce two objects: the fiberwise Hamiltonian skew-product and the sta-ble set at infinity for the Hamiltonian skew-product. These objects, for the minimal case such that ρ( ˜f ) has rational slope, are used to prove Theorem
Chapter 2
Notations and preliminaries
2.1
Maps, topological spaces and groups
Given a bijective map f : X → X, its set of fixed points will be denoted by Fix(f ). We shall write Per(f ) :=S
n≥1Fix(fn) for the set of periodic points.
When X is a topological space and A ⊂ X, we write int A for the interior of A and ¯A for its closure. When A is connected, we write cc(X, A) for the
connected component of X containing A.
When X is Hausdorff and locally-compact, we shall writeX := X t {∞}c
for its one-point compactification space.
If (X, d) is a metric space, the open ball of radius r > 0 and center x ∈ X will be denoted by Br(x). In this case, we also consider the space of compact
subsets
K(X) := {K ⊂ X : K is non-empty and compact}.
and we endow this space with its Hausdorff distance dH (induced by d)
defined by dH(K1, K2) := max ( sup p∈K1 inf q∈K2 d(p, q), sup q∈K2 inf p∈K1 d(p, q) ) , ∀K1, K2 ∈ K(X).
Given a locally compact non-compact topological space Y , we writeY :=b
Y t {∞} for the one-point compactification of Y . If A ⊂ Y is an arbitrary
subset,A will denote its closure inside the spaceb Y , and given any continuousb
proper map f : Y → Y , its unique extension to Y (that fixes the point atb
infinity) will be denoted by f :b Y →b Y .b
Whenever M1, M2, . . . Mndenote n arbitrary sets, we shall use the generic
notation pri: M1×M2×. . .×Mn→ Mito denote the ith-coordinate projection
Given any group G, we will write H < G when H is a subgroup of G and
H G when H is a normal subgroup.
2.2
Euclidean spaces and tori
We consider Rdendowed with its usual Euclidean structure denoted by h·, ·i. We write kvk := hv, vi1/2, for any v ∈ Rd. The unit (d − 1)-sphere is denoted
by Sd−1:= {v ∈ Rd: kvk = 1}. For any v ∈ Rd\ {0}, we define its orthogonal
complement Rv⊥ := {z ∈ Rd : hz, vi = 0} and, for each r ∈ R, the half-space Hvr :=
n
z ∈ Rd : hz, vi ≥ ro. (2.1)
For d = 2, given any v = (a, b) ∈ R2, we define v⊥ := (−b, a). Notice that in our notation, v⊥ belongs to the line Rv⊥.
We will also need the following notation for strips on R2: given v ∈ S1
and s > 0 we define the strip
Avs := H v −s∩ H −v −s = {z ∈ R2 : −s ≤ hz, vi ≤ s}. (2.2) Given any α ∈ Rd, T
α denotes the translation Tα: z 7→ z + α on Rd.
The d-dimensional torus Rd
/Zd will be denoted by Td and we write
π : Rd → Td for the natural quotient projection. Given any α ∈ Td, we
write Tα for the torus translation Tα: Td3 z 7→ z + α.
A point α ∈ Rd is said to be totally irrational when Tπ(α) is a minimal,
i.e. every orbit is dense in Td.
Making some abuse of notation, Lebdwill stand for the Lebesgue measure
on Rd as well as the Haar probability measure on Td.
2.3
Minimality
Let X be a topological space. A continuous map f : X → X is called minimal if X does not contain any non-empty, proper, closed f -invariant subset, or, equivalently, if the orbit of every point x ∈ X is dense in X.
A set A ⊂ Z is said to be relatively dense in Z when there is l ∈ N such that
A ∩ {n, n + 1, ..., n + l} 6= ∅
for each n ∈ Z.
Given a bijective map f : X → X, a point x ∈ X is said to be almost
fn(x) ∈ U } is relatively dense in Z. If X is compact and f is minimal, then
every point x ∈ X is almost periodic.
When X is a compact Hausdorff space, and f : X → X is minimal, then for every k ∈ N, there are a divisor l of k, l ≥ 1 and a partition {X0, ..., Xl−1}
of X in pairwise disjoint compact fk-minimal sets such that f (X
i) = Xi+1
for 0 ≤ i ≤ l − 2 and f (Xl−1) = X0. (for instance, see [Vri93])
So, the following result is a direct consequence of this fact.
Lemma 2.3.1. Let X be a connected and compact Hausdorff space. If a
map f : X → X is minimal, then f is totally minimal, i.e. fk is minimal for
every k ∈ N.
Remark 2.3.2. For any torus translation Tπ(α) and x ∈ Td, the closure O(x)
of the orbit of x is a finite union of tori of dimension k, with 0 ≤ k ≤ d, and the restriction of Tπ(α) to O(x) is minimal. (see [KH96]). So, every point in
O(x) is almost periodic by Tπ(α).
2.4
Surface topology
Let S be an arbitrary connected surface, i.e. a two-dimensional topological connected manifold.
An arc on S is a continuous map α : [0, 1] → S and a loop on S is a continuous map γ : T → S.
An open non-empty subset U ⊂ S is said to be inessential when every loop in U is contractible in S; otherwise it is said to be essential.
An open subset of S is said to be an (open) topological disk when it is homeomorphic to the open unitary disk {z ∈ R2 : kzk < 1}. Similarly, an
(open) topological annulus is an open subset of S homeomorphic to the open annulus T × R. Moreover, a subset A ⊂ S is said to be annular when it is an open topological annulus and none connected component of S \ A is inessential.
2.5
Groups of homeomorphisms
Given any topological manifold M , Homeo(M ) denotes the group of home-omorphisms from M onto itself. The subgroup formed by those homeomor-phisms which are homotopic to the identity map idM will be denoted by
Homeo0(M ).
We define the subgroup ^Homeo0(Td) < Homeo0(Rd) by
^ Homeo0(Td) := n ˜ f ∈ Homeo0(Rd) : ˜f − idRd ∈ C0(Td, Rd) o .
Notice that in this definition, we are identifying the elements of C0(Td, Rd) with those Zd-periodic functions from Rd to itself.
Making some abuse of notation, we also write π : ^Homeo0(Td) → Homeo0(Td)
for the map that associates to each ˜f the only torus homeomorphism π ˜f such
that ˜f is a lift of π ˜f .
Notice that with our notations, it holds πTα = Tπ(α) ∈ Homeo0(Td), for
every α ∈ Rd.
Given any ˜f ∈ ^Homeo0(Td), we define the displacement function by
∆f˜:= ˜f − idRd ∈ C0(Td, Rd). (2.3)
Observe that this function can be naturally considered as a cocycle over
f := π ˜f because ∆f˜n = n−1 X j=0 ∆f˜◦ fj, ∀n ≥ 1.
We write M(M ) for the space of Borel probability measures. For every µ ∈ M(M ) define
Homeoµ(M ) := {f ∈ Homeo(M ) : f?µ = µ} .
For any f ∈ Homeo(M ), let us write M(f ) := {ν ∈ M(M ) : f?ν = ν}.
In the particular case of d = 2, and for the sake of simplicity of nota-tion, we define the group of symplectomorphisms (also called area-preserving
homeomorphisms) by
Symp(T2) :=nf ∈ Homeo(T2) : Leb2 ∈ M(f )
o
.
It is well known that its connected component containing the identity, which will be denoted by Symp0(T2), coincides with Symp(T2)∩Homeo
0(T2).
We write
^
Symp0(T2) := π−1Symp0(T2)< ^Homeo0(T2).
2.6
Rotation set and rotation vectors
Let f ∈ Homeo0(Td) denote any homeomorphism and ˜f ∈ ^Homeo0(Td) be a
lift of f . We define the rotation set of ˜f by
ρ( ˜f ) := \ m≥0 [ n≥m ( ∆f˜n(z) n : z ∈ R d ) . (2.4)
Notice that ρ( ˜f ) is a compact connected subset of Rd. Notice that whenever ˜
f1, ˜f2 ∈ Homeo^0(Td) are such that π ˜f1 = π ˜f2, then π(ρ( ˜f1)) = π(ρ( ˜f2)).
Thus, given any f ∈ Homeo0(Td), we can just define
ρ(f ) := π(ρ( ˜f )) ⊂ Td. (2.5)
In particular, the fact of being a pseudo-rotation depends just on f and not on the chosen lift.
By (2.3), the rotation set is formed by accumulation points of Birkhoff averages of the displacement function, so given any µ ∈ M(f ) we can define the rotation vector of µ writing
ρµ( ˜f ) :=
Z
Td
∆f˜dµ, (2.6)
and it clearly holds ρµ( ˜f ) ∈ ρ( ˜f ).
When d = 2, after Misiurewicz and Ziemian [MZ89] we know that ρ( ˜f )
is not just connected but also convex. In fact, in the two-dimensional case it holds
ρ( ˜f ) =nρµ( ˜f ) : µ ∈ M
π ˜fo, ∀ ˜f ∈ ^Homeo0(T2). (2.7)
In [MZ89], it is proved the following property:
ρ( ˜fq− p) = qρ( ˜f ) − p, (2.8) when q ∈ N and p ∈ Z2.
2.7
Directional rotational deviations
Let f ∈ Homeo0(Td) be a periodic point free homeomorphism, and ˜f ∈
^
Homeo0(Td) be a lift of f . In such a case, from Handel’s result ([Han89]), we
know that there exist v ∈ Sd−1 and α ∈ R such that
ρ( ˜f ) ⊂ αv + Rv⊥.
With the notation introduced in 2.5, we can rewrite the definition of bounded v-deviations given in the Chapter 1. Then, we say that f exhibits
uniformly bounded directional rotational deviation (uniformly bounded v-deviations for short) when there exists M = M (f ) > 0 such that
D
∆f˜n(z), v E
− nα ≤ M, ∀z ∈ Td
Remark 2.7.1. Notice that the straight lines αv + Rv⊥ and (−α)(−v) + R(−v)⊥ coincide as subsets of R2. However à priori there is no obvious relation between v-deviation and (−v)-deviation.
Remark 2.7.2. Once again, notice that this concept of rotational deviation
does just depend on the torus homeomorphism and not on the chosen lift.
The next two sections give results that we will use later in our proofs.
2.8
Theorem of Gottschalk and Hedlund
This classical theorem guarantee the existence of a continuous solution of cohomological equation, under certain assumptions (see [Her79]).
Theorem ([GH55]). Let X a compact metric space, f : X → X a minimal
homeomorphism and φ : X → R a continuous function such that
sup n∈N n X i=0 φ(fi(x0)) < ∞
for some x0 ∈ X. Then, there is a continuous function u : X → R such that
u ◦ f − u = φ.
2.9
Theorem of Oxtoby and Ulam
Before we state the theorem, we will introduce the notion of OU-measures.
Definition 2. A Borel probability measure µ on a manifold X is called an
OU-measure if it is:
1. nonatomic i.e. it is zero on singleton sets; 2. positive on every nonempty open set; 3. zero on the manifold boundary.
Theorem ([OU41]). Let X be a compact connected n-manifold, with n ≥ 2.
If µ1 and µ2 are OU-measures on X such that µ1(X) = µ2(X), then there
exists a homeomorphism h : X → X such that h?µ1 = µ2.
In particular, if µ is an OU-measure in M(T2), there exists a
homeomor-phism h : T2 → T2 such that h
?µ = Leb2.
2.10
Hamiltonian homeomorphisms
In the symplectic setting, that is when ˜f ∈ ^Symp0(T2), the rotation vector of
Leb2 is also called flux of ˜f and it is usually denoted by Flux( ˜f ) = ρLeb2( ˜f ).
In this case, it can be easily shown that the flux map Flux : ^Symp0(T2) → R2
is indeed a group homomorphism. Since
Flux(Tp◦ ˜f ) = Tp(Flux( ˜f )), ∀p ∈ Z2, ∀f ∈ ^Symp0(T 2)
it one can clearly induce a map Symp0(T2) → T2, which, by some abuse of
notation, will be denoted by Flux, too.
The kernel of the group homomorphism Flux : Symp0(T2) → T2 shall be
denoted by
Ham(T2) := {f ∈ Symp0(T2) : Flux(f ) = 0} Symp0(T2), and their elements called Hamiltonian homeomorphisms.
Analogously, the kernel of the group homomorphism Flux : ^Symp0(T2) →
R2 is denoted by ] Ham(T2) := ˜ f ∈ ^Symp0(T2) : Flux( ˜f ) = 0 .
One can easily see that Ham(T2) and ]Ham(T2) can be naturally identified.
In fact, the restriction π
]
Ham(T2): ]Ham(T
2) → Ham(T2) is a topological group
isomorphism.
Now, notice that the following short exact sequence splits:
0 −→ Ham(T2) ,→ Symp0(T2)−−→ TFlux 2 −→ 0. (2.10) In fact, the map T2 3 α 7→ T
α is a section of Flux, and thus, the group
Symp0(T2) can be decomposed as a semi-direct product Symp
0(T2) = T2n
Ham(T2). In other words, given α, β ∈ T2 and h, g ∈ Ham(T2), we have (Tα◦ h) ◦ (Tβ ◦ g) = Tα+β ◦ (Adβ(h) ◦ g) , (2.11)
where we define Adβ(h) := Tβ−1◦ h ◦ Tβ ∈ Ham(T2).
This elementary fact about the group structure of Symp0(T2) will play a
key role in the contruction of the fiberwise Hamiltonian skew-product, the main new object of this work.
Chapter 3
Minimal Homeomorphisms
Assume that f ∈ Homeo0(T2) is a minimal homeomorphism. In this chapter,
we will give a description of its dynamics in the case that f exhibits uniformly bounded deviation.
We prove the equivalence between the uniform boundedness of rotational deviations and the existence of an f -invariant partition of T2. This partition
is called a pseudo-foliation, and its pseudo-leaves are connected essential sets with empty interior, which have the asymptotic direction parallel to the rotation set. The formal definition of pseudo-foliation will be given in 3.2.1. First, we need a technical result, which will be an important tool in our proofs.
3.1
A general result
The following theorem is about an arbtrirary minimal homeomorphism, with-out any assumption on its rotation set. We will use the next theorem to show a result about f with bounded deviation, and, in the next chapter, we will also use it to prove Theorem 1.0.1, assuming that f has unbounded deviation.
Consider the following notation: for p ∈ R2, denote the distance
d(p, Z2) := inf
q∈Z2kp − qk .
Theorem 3.1.1. Let f ∈ Homeo0(T2) be a minimal homeomorphism. Given
any v ∈ ρ( ˜f ) and > 0, there is δ > 0 such that, if n ∈ Z satisfies d(nv, Z2) <
δ, then there exists xn ∈ R2 with
˜ fn(x n) − xn− nv < .
Proof. Let µ ∈ M(f ) such that ρµ( ˜f ) = v. Since f is minimal, µ does not
in 2.9, there is a homeomorphism h ∈ Homeo0(T2) such that h?µ = Leb2.
As remarked in [KT14a], we can consider h isotopic to the identity, because
h can be chosen such that it lifts to a homeomorphism of R2 leaves the
boundary of [0, 1]2 pointwise fixed.
Define ˜g = ˜h ◦ ˜f ◦ ˜h−1. Then, g?Leb2 = Leb2 and Flux(˜g) = v.
Conse-quently, for all n ∈ Z, we have
Flux(Tv−n◦ ˜gn) = 0. Thus, T−n
v ◦ ˜gn∈ ]Ham(T2), and there exists zn∈ Fix(Tv−n◦ ˜gn) for each n.
Let xn = ˜h−1(zn) = zn+ ∆˜h−1(zn). We have:
˜
fn(xn) = ˜h−1(˜gn(zn)) = ˜h−1(zn+ nv) = zn+ nv + ∆˜h−1(zn+ nv)
= xn+ nv + ∆˜h−1(zn+ nv) − ∆˜h−1(zn)
Since ∆˜h−1 ∈ C0(T2, R2), given > 0, there exists δ > 0 such that, if
kp − qk < δ, we have
k∆˜h−1(p) − ∆˜h−1(q)k < .
Consider n ∈ Z such that nv is δ-close to vector of Z2, i.e., there exist
pn ∈ Z2 with knv − pnk < δ.
Remember that ∆h˜−1 is Z2-periodic. So,
k∆˜h−1(zn+ nv) − ∆˜h−1(zn)k = k∆˜h−1(zn+ nv) − ∆˜h−1(zn+ pn)k < . Therefore, we have ˜ fn(xn) − xn− nv = k∆˜h−1(zn+ nv) − ∆˜h−1(zn)k < .
Remark 3.1.2. Observe that in the proof, we only used the minimality of f
to guarantee that v is realized by a measure µ with full support. Therefore, Theorem 3.1.1 is still valid if one replaces the minimality hypothesis by f a homeomorphism of T2 homotopic to the identity and v = ρ
µ( ˜f ), where µ is
3.2
Bounded deviations
In this section we will study the following case: f minimal with bounded deviation. We show there is a pseudo-foliation of T2 invariant by f if and only if f exhibits uniformly bounded deviations. Furthermore, we consider in 3.2.2 the particular case with the rotation set of f contained in a line with rational slope. In this case, we prove that bounded deviation is equivalent to the existence of a semiconjugacy between f and some irrational rotation on the circle.
3.2.1
Pseudo-foliation
First, we will introduce the notion of pseudo-foliations.
Definition 3. A pseudo-foliation of R2 is a family F = {L
z}z∈R2 of closed
unbounded connected sets Lz, called pseudo-leaves, which is a partition of
the plane satisfying the following conditions: 1. Two distinct pseudo-leaves are disjoint; 2. Every pseudo-leaf has empty interior;
3. The complement of each pseudo-leaf Lzhas exactly two connected
com-ponents, which both are unbounded.
Definition 4. A pseudo-foliation of T2 is a family of connected sets which covers the torus, such that its lift to R2 is a pseudo-foliation of the plane, i.e.
a family
π(F ) = {Lπ(z):= π(Lz) : Lz ∈ F },
where F is a pseudo-foliation of R2.
We say that π(F ) is an f -invariant pseudo-foliation if f (Lπ(z)) = Lf (π(z)). Definition 5. Let π(F ) = {Lπ(z)} a pseudo-foliation of T2. We say that
π(F ) has the asymptotic direction of v if there is a pseudo-leaf Lz of its lift
contained in a strip bounded by two parallel straight lines, with the direction of v, such that both lines belong to different connected components of (Lz)C.
From the notation (2.2), there is s > 0 such that Lz ⊂ Avs.
Observe that, if π(F ) has the asymptotic direction of v, then every pseudo-leaf of its lift is contained in a strip bounded by two parallel straight lines, with the direction of v.
Remark 3.2.1. Recall that a topological foliation of T2 is a partition of the torus into one-dimensional topological submanifolds which is locally home-omorphic to the partition of the unit square by horizontal segments. So, any topological foliation is a particular case of pseudo-foliation of T2. More-over, Proposition 4.1 in [KK09] says that a topological foliation of T2 has an
asymptotic direction.
Now, we use the hypothesis of bounded deviation to construct a pseudo-foliation of T2 invariant by f . This construction will be a consequence of
Gottschalk-Hedlund Theorem (see Section 2.8).
For this, we will use the following result:
Lemma 3.2.2. If f has unbounded v-deviation, then f also has unbounded
(−v)-deviation.
This result will be proved in Chapter 4, Lemma 4.1.4. Although we will show this result for the particular case v = (1, 0), the same idea proves Lemma 3.2.2, for v in general.
Assume that f exhibits uniformly bounded v-deviations. Then, there is
M = M (f ) > 0 such that
D
∆f˜n(z), v E
− nα ≤ M,
for all z ∈ T2 and n ∈ Z.
Consider φ : T2 → R given by φ(z) =D∆ ˜ f(z), v E − α. Then, n−1 X j=0 φ( ˜fj(z)) =D∆f˜n(z), v E − nα ≤ M.
From Lemma 3.2.2, we have that there is M > 0 such that for all z ∈ T2
and n ∈ Z, n−1 X j=0 φ( ˜fj(z)) ≤ M.
So, the Gottschalk-Hedlund Theorem implies that there is a continuous function u : T2 → R such that u ◦ ˜f − u = φ. So,
u( ˜f (z)) − u(z) =D∆f˜(z), v
E
− α =Df (z) − z, v˜ E− α.
Let ˜H : R2 → R,
˜
H(z) = hz, vi − u(z). (3.1) The equality above implies that
˜
H( ˜f (z)) = ˜H(z) + α.
Remark 3.2.3. Notice that, since u is Z2-periodic, then u(z + k) = u(z) for
every k ∈ Z2. So, ˜
H(z + k) = hz + k, vi − u(z + k) = ˜H(z) + hk, vi .
Thus, if r ∈ R and k ∈ Z2, it holds: ˜H−1(r) + k = ˜H−1(r + hk, vi). The next results describe some properties ofH˜−1(r)
r∈R. These
proper-ties will imply the set of connected components of H˜−1(r)
r∈R satisfies the
conditions of Definition 3, i.e. this set is a pseudo-foliation of R2.
Remember that, for all z ∈ R2 and n ∈ N, we have ˜H( ˜fn(z)) = ˜H(z)+nα.
So, if r ∈ R and z ∈ ˜H−1(r), then ˜fn(z) ∈ ˜H−1(r + nα). Therefore, for all
n ∈ N it holds:
˜
fn( ˜H−1(r)) = ˜H−1(r + nα), and we have that the set H˜−1(r)
r∈R is ˜f -invariant.
Remark 3.2.4. Let r ∈ R and Mu = max |u(z)|, where u satisfies (3.1). Notice
that, for every z ∈ ˜H−1(r), we have
| hz, vi − r| = |u(z)| ≤ Mu.
This implies that ˜H−1(r) is contained in a strip Sr bounded by two parallel
lines with the direction of v⊥. Moreover, each strip Sr has width equal 2Mu
(independent of r).
The following result states that any connected component of ˜H−1(r) is an unbounded set of R2 as required on Definition 3.
Lemma 3.2.5. For each r ∈ R, every connected component of ˜H−1(r) is
Proof. First, we will prove that ˜H−1(r) is unbounded in both directions. Indeed, from Remark 3.2.4, there exists a strip Sr with the direction of v⊥
and width equal 2Mu such that ˜H−1(r) ⊂ Sr. Let z ∈ ˜H−1(r) and consider
λ > 0 such that λ kvk2 > 2Mu and the points z + λv and z − λv are contained
in the complementar of Sr.
Let γ : [0, 1] → R2 an arbitrary curve which satisfies γ(0) = z − λv and
γ(1) = z + λv. Therefore, we have: ˜ H(γ(0)) = H(z − λv) = ˜˜ H(z) − λ kvk2+ u(z) − u(z − λv) ≤ r − λ kvk2+ 2Mu < r ; ˜ H(γ(1)) = H(z + λv) = ˜˜ H(z) + λ kvk2+ u(z) − u(z + λv) ≥ r + λ kvk2− 2Mu > r.
So, by the Intermediate Value Theorem applied to ˜H ◦ γ, there is t0 ∈
(0, 1) with ˜H(γ(t0)) = r and we have γ(t0) ∈ ˜H−1(r). This implies that
˜
H−1(r) separates the points γ(0) and γ(1). Since these points in the plane are separated by the closed set ˜H−1(r), then there is some connected component of ˜H−1(r) that separates these points (see [New92]).
Then, ˜H−1(r) has an unbounded (in both directions) connected compo-nent.
It remains to prove that it is true for every connected component of ˜
H−1(r). In fact, let z be a point in ˜H−1(r) and Γ be a connected component of ˜H−1(r) containing z. Let us suppose Γ is bounded.
We know there exists an unbounded connected component ξ of ˜H−1(0), and let w be an arbitrary point in ξ. Since f is minimal, there is a sequence
nj ∈ N such that
fnj(π(w)) → π(z) (3.2)
We will assume without loss of generality that v⊥ has a nonzero second coordinate. Let A = T × R be the annulus obtained as the quotient space R2/T(1,0) and let ¯f : A → A be a lift of f . We denote A := Rb 2t {−∞, ∞}
the two-points compactification of A, and f :b A →b A the unique continuousb
extension of ¯f that fixes the points ±∞.
In the compactification A, considerb ξ the closure of the projection of ξb
in A. We have that fbnj(ξ)b
is a sequence of compact sets of A containingb
the points −∞ and ∞. So, it has a convergent subsequence in the Hausdorff topology to a compact set θ.b
Notice that {−∞, ∞} ⊂θ. Considering θ ⊂ Rb 2 a correspondent set of θ,b
this implies that θ is an unbounded set in both vertical directions. On the other hand, considering the projection π(θ) ⊂ T2, from (3.2), we have:
π(z) ∈ π(θ).
So, z + k ∈ θ for some k ∈ Z2. Thus θ ⊂ ˜H−1(r) + k. Since we have
z + k ∈ θ ∩ (Γ + k),
then θ ⊂ Γ + k. This contradicts our assumption that Γ is bounded.
Note that, by definition, we have ˜
H−1(r) ∩ ˜H−1(s) = ∅,
whenever r 6= s. Then, the first condition of Definition 3 clearly holds for the set of connected components of H˜−1(r)
r∈R.
The following result shows this set also satisfies the second condition.
Lemma 3.2.6. For each r ∈ R, ˜H−1(r) has empty interior.
Proof. Let r be an arbitrary fixed real number, and suppose that ˜H−1(r) has non-empty interior. So, we have an open set U of R2 such that U ⊂ ˜H−1(r). Since f is minimal, we know that
π [ n≥0 ˜ fn( ˜H−1(s)) = T2,
for every s ∈ R. So, there are ns ∈ N and ks∈ Z2 such that
˜
fns( ˜H−1(s)) ∩ (U + k
s) 6= ∅.
From Remark 3.2.3, we have
U + ks⊂ ˜H−1(r) + ks = ˜H−1(r + hks, vi).
Since ˜fns( ˜H−1(s)) = ˜H−1(s + n
sα), we have
˜
H−1(s + nsα) ∩ ˜H−1(r + hks, vi) 6= ∅.
Since pre-image of distinct numbers are disjoint sets, then we conclude that, for every s ∈ R, there are ns ∈ N and ks∈ Z2 such that
which is absurd. In fact, since
Cr := {r + hk, vi − nα; k ∈ Z2 , n ∈ Z}
is an enumerable subset of R, there is s0 ∈ C/ r.
To see that we have a pseudo-foliation of R2, we need to verify the third condition of Definition 3, i.e. we need to show that every connected com-ponent of ˜H−1(r) separates the plane in exactly two connected components. This is a consequence of Lemma 3.2.5.
Lemma 3.2.7. For each r ∈ R, the complement of every connected
com-ponent of ˜H−1(r) has exactly two connected components, which both are
un-bounded.
Proof. From Lemma 3.2.5, we know that every connected component of
˜
H−1(r) is unbounded (in both directions), and the complement has at least two connected components.
If we have a connected component Γ of ˜H−1(r) such that its comple-ment has more than two connected components, then there exists Ω0 ⊂ Sr a
connected component of ΓC which satisfies one of the following cases:
1. Ω0 is bounded from above or below;
2. Ω0 is unbounded in both directions.
Suppose that we have the first case. Since ˜H−1(r) has empty interior, there is w ∈ Ω0 such that ˜H(w) = s 6= r. Consider θ the connected
com-ponent of ˜H−1(s) containing w. Lemma 3.2.5 implies that θ intersects the boundary of Ω0. Thus, θ ∩ Γ 6= ∅, a contradiction.
Now, suppose that we have the second case. Denote by l1 and l2 two
par-allel straight lines, with the direction of v⊥, such that both lines belong to different connected components of ΓC. Let Ω
1 and Ω2 be the connected
com-ponents of ΓC containing l1 and l2 respectively. Consider w ∈ Ω0 such that
˜
H(w) = s 6= r and let θ be the connected component of ˜H−1(s) containing
w. Then, θ ⊂ Ω0.
Notice that l1 and l2 belong to different connected components Σ1 and Σ2
of θC. Indeed, if it is not true, then given two points z
1 ∈ l1 and z2 ∈ l2, there
exists a curve γ : [0, 1] → R2 such that γ(0) = z
1, γ(1) = z2 and γ(t) /∈ θ, for
all t ∈ [0, 1]. But it implies that θ is bounded from above or below, which is a contradiction with Lemma 3.2.5.
Since Γ ∩ θ = ∅, we have Γ ⊂ θC. Observe at least one of the components Σi does not intersect Γ, for i = 1, 2. Consider a curve γ : [0, 1] → R2 such
that γ(0) ∈ li, γ(1) ∈ θ and γ(t) ∈ Σi , for all t ∈ [0, 1). Since θ ⊂ Ω0 and li
is contained at ΩC0, we have γ ∩ Γ 6= ∅. But it contradicts that Γ and Σi are
disjoint sets.
Then, we conclude that ΓC has exactly two connected components.
Therefore, the lemmas above imply that the set of connected components of pre-images H˜−1(r), considering all r ∈ R, is a pseudo-foliation of R2 invariant by ˜f , which is a lift of a pseudo-foliation of T2 invariant by f . So,
we proved the following result:
Proposition 3.2.8. Let f be a minimal homeomorphism homotopic to the
identity with ρ( ˜f ) ⊂ αv+Rv⊥, with v ∈ R2. Suppose that f exhibits uniformly bounded v-deviations. Then there is an f -invariant pseudo-foliation which has the asymptotic direction of v⊥.
Remark 3.2.9. Note that, if v ∈ Z2, the strip S
r defined in Remark 3.2.4 is
bounded by two parallel lines with the direction of v⊥ ∈ Z2. So, π(S
r) is a
compact set of T2. Therefore, in this case, the f -invariant pseudo-foliation
of T2 constructed above has compact pseudo-leaves.
It is a natural question to ask whether the converse of Proposition 3.2.8 is also true. If f is minimal, with ρ( ˜f ) ⊂ αv + Rv⊥, such that there is an
f -invariant pseudo-foliation with the asymptotic direction of v⊥, then are the
v-deviations bounded?
If f is a minimal homeomorphism homotopic to the identity such that the rotation set is a segment with rational slope, the question above has a positive answer, i.e. f has bounded v-deviations. In fact, by Theorem 1.0.1, which will be proved in the next chapter, if f has unbounded deviation, then
f is spreading. But, from the proof of Proposition 4.2 in [KK09], f spreading
implies that f has no invariant pseudo-foliation with the direction of ρ( ˜f ).
Now, we will use Theorem 3.1.1 and the idea of [KK09] to show the converse of Proposition 3.2.8 is valid in more general case:
Theorem 3.2.10. Let f be a minimal homeomorphism homotopic to the
identity with ρ( ˜f ) ⊂ αv + Rv⊥, with v ∈ R2. Then the following statements are equivalent:
1. f exhibits uniformly bounded v-deviations;
2. there is an f -invariant pseudo-foliation which has the asymptotic di-rection of v⊥.
Proof. By Proposition 3.2.8, we have that (1) implies (2).
Suppose that f satisfies (2). We will show that f satisfies (1), i.e. f has bounded v-deviations. We will assume without loss of generality that v is a unit vector, which has a nonzero second coordinate.
Consider L a pseudo-leaf of the ˜f -invariant pseudo-foliation of R2, with
asymptotic direction of v⊥. Denote S the strip bounded by L and L + (0, 1).
Note that we have
[
n∈Z
S + (0, n) = R2.
Since the pseudo-foliation has the direction of v⊥, then the orthogonal projection of S on the direction of v is a segment with bounded width. Let
W be this width.
On the other hand, for each n ∈ Z, there exists m ∈ Z such that ˜
fn(L) ⊂ S + (0, m).
So, ˜fn(L + (0, 1)) ⊂ S + (0, m + 1). Thus, ˜fn(S) is contained in the strip bounded by L + (0, m) and L + (0, m + 2). This implies that the orthogonal projection of ˜fn(S) on the direction of v has width less or equal than 2W ,
for every n ∈ Z.
Now, we will use Theorem 3.1.1 to see that the bounded width of ˜fn(S)
implies bounded deviation. Let u = αv + tv⊥ ∈ ρ( ˜f ). We have that, given > 0, there is δ > 0 such that, if n ∈ Z satisfies d(nu, Z2) < δ, then there exists xn∈ R2 with ˜ fn(xn) − xn− nu < .
Consider (nj)j∈N a sequence such that d(nju, Z2) < δ. By Remark 2.3.2,
there exists l ∈ N with 0 < nj+1− nj ≤ l for every j ∈ N. We will assume
that xnj ∈ S for every j ∈ N.
Let z ∈ R2. Since the difference ˜fnj(z) − z is Z2-periodic, translating by
integer vector if necessary, we can assume that z ∈ S. Then we have
D ˜ fnj(z) − z, vE− n jα = D ˜ fnj(z) − ˜fnj(x nj), v E −Dz − xnj, v E + Df˜nj(x nj) − xnj− nju, v E .
Notice that Df˜nj(z) − ˜fnj(x
nj), v E
is less or equal than the width of the orthogonal projection of ˜fnj(S) on the direction of v, i.e. ,
D ˜ fnj(z) − ˜fnj(x nj), v E ≤ 2W. Furthermore, −Dz − xnj, v E ≤ W and D ˜ fnj(x nj) − xnj− nju, v E ≤ f˜ nj(x nj) − xnj − nju < .
Thus, for every j ∈ N and z ∈ R2,
D
˜
fnj(z) − z, vE− n
jα < 3W + (3.3)
For nj ≤ n ≤ nj+1, n = nj + i with 1 ≤ i ≤ l, we have:
D ˜ fn(z) − z, vE− nα = Df˜nj( ˜fi(z)) − z, vE− (n j+ i)α = Df˜nj( ˜fi(z)) − ˜fi(z), vE− n jα + D ˜ fi(z) − z, vE− iα From (3.3), Df˜nj( ˜fi(z)) − ˜fi(z), vE− n jα < 3W + . Moreover, we have D ˜ fi(z) − z, vE− iα ≤ f˜ i (z) − z − iα < max ∆f˜l(w) . Therefore, D ˜ fn(z) − z, vE− nα < 3W + + max ∆f˜l(w) . Denote M = M (f ) = 3W + + max ∆f˜l(w)
. So, we showed that D
∆f˜n(z), v E
− nα ≤ M, for all z ∈ T2 and n ∈ Z, proving that f satisfies (1).
3.2.2
Semiconjugacy
From now on, we will assume that f is minimal with bounded deviation and rotation set contained in a line with rational slope. In other words, we consider ρ( ˜f ) ⊂ αv + Rv⊥, with v ∈ Z2. Notice that we have ρ( ˜f ) ∩ Q2 = ∅.
Then, it implies α ∈ R \ Q.
We will show that f is semiconjugate to Rα. Moreover, we will prove that
the existence of this semiconjugacy is a sufficient condition to guarantee that the deviation is bounded.
Proposition 3.2.11. Let f be a minimal homeomorphism homotopic to the
identity with ρ( ˜f ) ⊂ αv + Rv⊥, with v ∈ Z2. Then f exhibits uniformly
bounded v-deviations if and only if f is semiconjugate to the irrational rota-tion Rα on T.
Proof. Suppose that f exhibits uniformly bounded v-deviations. Let ˜H be
the map given by (3.1). In Section 3.2.1, we showed that ˜
H(z + k) = hz + k, vi − u(z + k) = ˜H(z) + hk, vi ,
for every k ∈ Z2.
Since we are considering v ∈ Z2, then hk, vi ∈ Z. Thus, we have H :
T2 → T such that ˜H is a lift of H and satisfies H ◦ f = Rα◦ H. We conclude
that H is a semiconjugacy between f and Rα.
Now, suppose that f is semiconjugate to the irrational rotation Rα on
T.
Jäger and Passeggi ([JP13]) proved that, if f is semiconjugate to the irrational rotation Rα on T, then there exists a semiconjugacy h from f to
Rα such that all fibres are essential annular continua.
Considering ˜h the lift of h such that ˜h ◦ ˜f = Rα◦ ˜h, this implies that the
pre-image sets ˜h−1(r) form a partition of the plane such that the complement of each ˜h−1(r) has 2 unbounded connected components. A similar argument to the proof of Lemma 3.2.6, using the minimality of f , implies that ˜h−1(r) has empty interior. Thus, we have an f -invariant pseudo-foliation with the pre-image sets ˜h−1(r).
Therefore, Theorem 3.2.10 implies that f has bounded v-deviations.
Remark 3.2.12. In the reciprocal of Proposition 3.2.11, we used the result of
Jäger and Passeggi assuming that f is minimal, but their result is stronger. They proved this fact for f ∈ Homeo0(T2) in general, without the hypothesis
of minimality (See Theorem 1 of [JP13]).
Remark 3.2.13. Proposition 3.2.11 is valid for a more general case. Jäger and
Tal([JT14]) proved this result for a non-annular area-preserving homeomor-phism.
Chapter 4
Unbounded deviations
In this chapter we will introduce new objects, which will be used to prove Theorem 1.0.1. These objects and the techniques of our initial proofs are also valid for the more general case of periodic point free homeomorphisms. Then, in the following section, we will consider f ∈ Symp0(T2), without
periodic points and with unbounded deviation.
4.1
The fiberwise Hamiltonian skew-product
and the stable set at infinity
The objects, which will be introduced in this section, are inspired in the sets
ω± used in [GKT12]. These sets were motivated by Birkhoff’s works, and
they were used in many papers, for example: [AZT11], [Tal12], [BT13] and [KT14b].
Then, before we define our objects, we will give a short introduction about these sets ω± and explain how they motivated the techniques of our proofs.
4.1.1
Motivation
From (2.1), we will use the following notation for the half-spaces: H±(1,0)0 :=
n
z ∈ R2 : hz, ±(1, 0)i ≥ 0o=nz ∈ R2 : ±pr1(z) ≥ 0o; (4.1) Define the sets ω± ⊂ R2 as the union of all unbounded connected
com-ponents of
\
n∈Z
˜
fn(H±(1,0)0 ).
We have the following properties: ω± are ˜f -invariants; every connected
of ω− is unbounded to the left. In [GKT12], the authors considered ρ( ˜f ) =
{0} × [a, b], with a < 0 < b. In this case, the sets ω± are non-empty. This
fact is used to prove that ˜f has bounded (1, 0)-deviation.
If ρ( ˜f ) is the vertical segment ρ( ˜f ) = {α} × [a, b], with α ∈ R \ Q and a < 0 < b, we have ω±= ∅. So, for the case that we will consider, these sets
are not useful.
If we consider the analogous set as the union of all unbounded connected components of
\
n∈Z
(T(α,0)−n ◦ ˜fn)(H±(1,0)0 ),
then these sets are non-empty. But we lose the property of invariance by ˜f .
Again, these sets are not useful to our proofs.
Thus, the idea will be used here is the following: we will consider a map
F : T2×R2 → T2×R2 such that F is a skew-product, which is the translation
T(α,0) on the first coordinate, such that
Fn(0, z) =n(α, 0), ˜fn(z) − n(α, 0).
So, we have ρ(F ) ⊂ {(α, 0)} × ({0} × R). For this map, analogously to the sets ω±, we will define, for each fiber t ∈ T2, the set Λ±r(t) as the union of all
unbounded connected components of
{t} × R2∩ \ n∈Z Fn(T2× H±(1,0)r ) ,
and the set Λ±r will be the union of Λ±r(t) considering all t ∈ T2. We will
prove that Λ±r 6= ∅. Moreover, F maps Λ±
r(t) to Λ±r(t + (α, 0)), and this
implies that Λ±
r is F -invariant.
In the following, we will give the formal definitions of the skew-product and the sets Λ±r. First, the definitions will be given for the more general case with ρ( ˜f ) ⊂ αv + Rv⊥, v ∈ R2. Later, we will use these definitions for the
case cited above, with a vertical segment as the rotation set.
4.1.2
Definition and properties
Given any f ∈ Symp0(T2), let ˜f ∈ ^Symp0(T2) denote an arbitrary but fixed lift of f . For simpicity, let us just write ρ := Flux(f ) ∈ T2and ˜ρ := Flux( ˜f ) ∈
R2.
Inspired by (2.11), we define a continuous map H : T2 → ]Ham(T2) by
Ht := π−1 Adt Tρ−1◦ f= π−1Tt−1◦ Tρ−1◦ f ◦ Tt , ∀t ∈ T2, (4.2)
recalling that π
]
Ham(T2): ]Ham(T
2
) → Ham(T2) is a topological group isomor-phism.
Considering H as a cocycle over the torus translation Tρ: T2 → T2, one
defines the skew-product homeomorphism F : T2× R2
→ T2
× R2 by
F (t, z) := (Tρ(t), Ht(z)) , ∀(t, z) ∈ T2× R2,
and one can easily show that
F (t, z) =t + ρ, z + ∆f˜
t + π(z)− ˜ρ, ∀(t, z) ∈ T2 × R2, (4.3) where ∆f˜∈ C0(T2, R2) is the displacement function given by (2.3). We will
use the following classical notation for cocycles: given n ∈ Z and t ∈ T2, we write Ht(n):= idT2, if n = 0; Ht+(n−1)ρ◦ Ht+(n−2)ρ◦ · · · ◦ Ht, if n > 0; Ht+nρ−1 ◦ · · · ◦ Ht−2ρ−1 ◦ Ht−ρ−1, if n < 0. With such a notation it holds Fn(t, z) := Tn
ρ(t), H (n) t (z) , for all (t, z) ∈ T2× R2 and every n ∈ Z.
This skew-product F will play a fundamental role in our analysis of rota-tional deviations and the following simple formula for iterates of F represents the main reason:
Fn(t, z) = Tρn(t), Ht+(n−1)ρ Ht+(n−2)ρ . . . Ht(z) !! = t + nρ, π−1 Adt+(n−1)ρTρ−1◦ f◦ . . . ◦ Adt Tρ−1◦ f (z) = t + nρ, π−1 Adt Tρ−n◦ fn (z) , (4.4) for every (t, z) ∈ T2× R2 and every n ∈ N.
From now on, we shall assume f is periodic point free, α ∈ R and v ∈ S1
such that
ρ( ˜f ) ⊂ αv + Rv⊥. (4.5) Notice that in such a case, h ˜ρ, vi = α.
From (4.4) it easily follows that a point z ∈ R2 exhibits bounded v-deviations (as in (1.1)) if and only if
D
