# A priori bounds for GIETs, affine shadows and rigidity of foliations in
genus two

Selim Ghazouani  and  Corinna Ulcigrai

###### Abstract.

We prove a rigidity result for foliations on surfaces of genus two, which can
be seen as a generalization to higher genus of Herman’s theorem on circle
diffeomorphisms and, correspondingly, flows on the torus. We prove in
particular that, if a smooth, orientable foliation with non-degenerate (Morse)
singularities on a closed surface of genus two is minimal, then, under a full
measure condition for the rotation number, it is _differentiably_ conjugate to
a _linear_ foliation.

The corresponding result at the level of Poincaré sections is that, for a full
measure set of (standard) interval exchange transformations (IETs for short)
with $d=4$ or $d=5$ continuity intervals and irreducible combinatorics, any
generalized interval exchange transformation (GIET for short) which is
topologically conjugate to a standard IET from this set and satisfies an
obstruction expressed in terms of boundary operator (which is automatically
satisfied when the GIET arises as a Poincaré map of a smooth foliation) is
$\mathcal{C}^{1}$-conjugate to it. This in particular settles a conjecture by
Marmi, Moussa and Yoccoz in genus two. Our results also show that this
conjecture on the rigidity of GIETs can be reduced to the study of affine
IETs, or more precisely of Birkhoff sums of piecewise constant observables
over standard IETs, in genus $g\geq 3$.

Our approach is via renormalization, namely we exploit a suitable acceleration
of the Rauzy-Veech induction (an acceleration which makes Oseledets generic
_effective_) on the space of GIETs. For infinitely renormalizable, irrational
GIETs of _any_ number of intervals $d\geq 2$ we prove a dynamical dichotomy on
the behaviour of the orbits under renormalization, by proving that either an
orbit is _recurrent_ to certain bounded sets in the space of GIETs, or it
diverges and it is approximated (up to lower order terms) by the orbit of an
affine IET (a case that we refer to as _affine shadowing_). This result can in
particular be used, in conjunction with previous work by Marmi-Moussa and
Yoccoz on the existence of wandering intervals for affine IETs, to prove, _a
priori bounds_ in genus two and is therefore at the base of the rigidity
result.

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Geometric rigidity in dynamics.
    2. 1.2 Geometric rigidity in genus two
    3. 1.3 Diffeomorphisms of the circle and foliations on the torus.
    4. 1.4 Flows in higher genus and interval exchange transformations.
      1. 1.4.1 Renormalization and combinatorics.
      2. 1.4.2 Absence of a Denjoy theorem and wandering intervals.
      3. 1.4.3 Cohomological equations and obstructions to linearisation.
      4. 1.4.4 Rigidity conjecture.
    5. 1.5 Rigidity result for GIETs
      1. 1.5.1 Regularity of the conjugacy
      2. 1.5.2 The boundary assumption
      3. 1.5.3 The $\mathcal{C}^{0}$-conjugacy class in parameter space
    6. 1.6 A dynamical renormalization dichotomy and the strategy of the proof.
      1. 1.6.1 Wandering intervals and a priori bounds
      2. 1.6.2 Boundary obstructions and convergence of renormalization
      3. 1.6.3 The Diophantine-type condition
    7. 1.7 Organization of the paper and reading guide.
  2. 2 Background material
    1. 2.1 Interval exchange transformations
      1. 2.1.1 Generalized interval exchange transformations
      2. 2.1.2 Standard, affine and Moebius IETs
      3. 2.1.3 Combinatorial data
      4. 2.1.4 Singularities
      5. 2.1.5 Connections
      6. 2.1.6 GIETs, surfaces and foliations
    2. 2.2 Parameter (sub)spaces
      1. 2.2.1 Parameter spaces of IETs
      2. 2.2.2 Parameters of AIETs
      3. 2.2.3 Shape-profile coordinates for GIETs
    3. 2.3 Renormalization of GIETs
      1. 2.3.1 Elementary step of Rauzy-Veech induction
      2. 2.3.2 Paths on Rauzy diagrams and rotation numbers
      3. 2.3.3 Irrational combinatorial rotation numbers and semi-conjugacy with a standard IET
      4. 2.3.4 Periodic-type (or Fibonacci-type) combinatorics
      5. 2.3.5 Definition of the renormalisation operator
      6. 2.3.6 Accelerations
      7. 2.3.7 Dynamical partitions
    4. 2.4 One dimensional dynamics toolkit
      1. 2.4.1 Non-linearity
      2. 2.4.2 Distortion bounds
      3. 2.4.3 The Schwarzian derivative
    5. 2.5 The Zorich cocycle
      1. 2.5.1 Invariant measures
      2. 2.5.2 Natural extension.
      3. 2.5.3 Basics on cocycles.
      4. 2.5.4 Induced cocycles and accelerations.
      5. 2.5.5 Integrability
      6. 2.5.6 The Zorich cocycle.
      7. 2.5.7 Dynamical interpretation of the entries.
      8. 2.5.8 Length cocycle
      9. 2.5.9 Cocycle action on log-slopes of AIETs
      10. 2.5.10 Lyapunov exponents and Oseledets splittings
    6. 2.6 Birkhoff sums and special Birkhoff sums
      1. 2.6.1 Piecewise continuous functions and Birkhoff sums
      2. 2.6.2 Special Birkhoff sums
      3. 2.6.3 The extended Zorich cocycle
      4. 2.6.4 Decomposition of Birkhoff sums
    7. 2.7 Boundary operators
      1. 2.7.1 Boundary operator for observables
      2. 2.7.2 Cohomological interpretation of the boundary operator
      3. 2.7.3 Renormalization invariance of the boundary operator for GIETs
      4. 2.7.4 Boundary of a GIET
  3. 3 Affine shadowing
    1. 3.1 Scaling invariants and mean (log-)slope vectors
    2. 3.2 Affine shadowing in the periodic type (or Fibonacci type) case
    3. 3.3 The Diophantine-type condition for the general case
      1. 3.3.1 Notion of full measure
      2. 3.3.2 Oseledets generic extensions
      3. 3.3.3 Good return times
      4. 3.3.4 Notation for Zorich accelerations
      5. 3.3.5 The Regular Diophantine Condition
    4. 3.4 Affine Shadowing under the Regular Diophantine Condition
    5. 3.5 The affine shadowing dichotomy
      1. 3.5.1 The shadowing lemma, general case
      2. 3.5.2 Linear approximation error estimate
      3. 3.5.3 Building the shadow
      4. 3.5.4 Control of the unstable part.
    6. 3.6 Control of the central part
    7. 3.7 Control of the stable part
    8. 3.8 Proof of Theorem 3.2
  4. 4 Convergence of renormalization in the recurrent case
    1. 4.1 Outline of the proof.
    2. 4.2 Preliminaries: distances and a priori bounds
      1. 4.2.1 Distances on parameter space
      2. 4.2.2 $\mathcal{C}^{k}$-bounded sets
      3. 4.2.3 Distances comparision
      4. 4.2.4 Boundary stratification and regimes
      5. 4.2.5 A priori bounds.
    3. 4.3 Exponential decay of the dynamical partitions mesh.
      1. 4.3.1 Balance of continuity intervals via good $\mathcal{C}^{1}$-recurrence sequence
      2. 4.3.2 Decay of the mesh at good $\mathcal{C}^{1}$-recurrence times.
    4. 4.4 Convergence to Moebius maps
      1. 4.4.1 Estimates of the distance from MIETs.
      2. 4.4.2 Exponential convergence to MIETs.
      3. 4.4.3 Parameters of MIETs
    5. 4.5 Convergence to AIETs
      1. 4.5.1 A combinatorial lemma
      2. 4.5.2 Non-linearity decrease at $\mathcal{C}^{1}$\- recurrence times
      3. 4.5.3 Exponential decay of the total non-linearity
    6. 4.6 Convergence to IETs
  5. 5 Rigidity for GIETs
    1. 5.1 Regularity of the conjugacy
    2. 5.2 Wandering intervals and distorted towers
      1. 5.2.1 Distorted towers
      2. 5.2.2 Exponentially distorted towers
      3. 5.2.3 Reduction to the affine shadow.
    3. 5.3 Towers distorsion via Birkhoff sums
      1. 5.3.1 Partition size estimates via Birkhoff sums.
      2. 5.3.2 Wandering intervals in affine IETs.
      3. 5.3.3 Proof of Proposition 5.2.1
    4. 5.4 Final arguments in the proof of Theorem 5.1
    5. 5.5 A priori bounds in genus two.
  6. 6 Rigidity of foliations.
    1. 6.1 Preliminaries on singular foliations and holonomy
      1. 6.1.1 Foliations singularities and regularity
      2. 6.1.2 Holonomy around singular points
      3. 6.1.3 Minimality (or quasi-minimality) for a foliation
      4. 6.1.4 Relation with GIETs
    2. 6.2 Conjugacies of foliations and rigidity
      1. 6.2.1 Linear foliations
      2. 6.2.2 Linearization of minimal foliations
      3. 6.2.3 Measure (class) on minimal foliations
      4. 6.2.4 Rigidity of foliations in genus two.
  7. 7 Full measure of the Regular Diophantine Condition
    1. 7.1 Full measure of Conditions $(i)$ and $(ii)$
      1. 7.1.1 Full measure of generic Oseledets extensions
      2. 7.1.2 Construction of good return times
    2. 7.2 Effective Oseledets
      1. 7.2.1 Effective Oseledets return times
      2. 7.2.2 Construction of effective Oseledets sequences
    3. 7.3 Control of the series in Condition $(iii)$ and proof of full measure.
      1. 7.3.1 Induced cocycle
      2. 7.3.2 Acknowledgements
  8. A
    1. A.1 Boundary and suspensions.
    2. A.2 Distorsion bounds for GIETs
    3. A.3 Distances comparisons.
      1. A.3.1 The semi-distance $d_{\eta}$ is a distance.
      2. A.3.2 Comparison of $d_{\eta}$ and $d_{\mathcal{C}^{1}}$ on $\mathrm{Diff}^{1}([0,1])$.
      3. A.3.3 Comparison of $d_{\eta}$ and $d_{\mathcal{C}^{1}}$ distances from AIETs.
      4. A.3.4 The Schwarzian derivative and the $\mathcal{C}^{3}$-distance
    4. A.4 Lipschitz regularity of composition and renormalization
      1. A.4.1 Lipschitz regularity of composition
      2. A.4.2 Lipschitz regularity of renormalization
    5. A.5 Saddles defined by a non-degenerate vector field

##  1\. Introduction and main results

In this article we extend some aspects of the theory of _circle
diffeomorphisms_ and of _flows on the torus_ to the context of generalized
interval exchange transformations and of foliations on higher genus surfaces.
Exploiting a renormalization approach, we are able in particular to prove a
_rigidity_ result in genus two which can be seen as a generalization of a
celebrated theorem by Herman in genus one and proves a conjecture by Marmi,
Moussa and Yoccoz in [MMY3] in the case of generalized interval exchange
transformations which correspond to minimal surface flows in genus two. We
start by giving an introduction to geometric rigidity problems in dynamics and
some key results on circle diffeomorphisms and (generalized) interval exchange
transformations.

###  1.1. Geometric rigidity in dynamics.

A natural problem in the theory of smooth dynamical systems is to establish
which classes of dynamical systems are geometrically rigid in the following
sense. We say that a class of dynamical systems (whether it be an endomorphism
of a manifold, a foliation or a flow) is geometrically rigid (or just rigid)
if a topological conjugacy (namely a homeomorphism which intertwines the
dynamics on the two systems, see below for the definition) between two
elements in this class is necessarily differentiable. A natural problem in the
theory of smooth dynamical systems is to establish which classes of dynamical
systems are geometrically rigid.

It is well-known that periodic orbits provide obstructions to geometric
rigidity for _hyperbolic dynamical systems_ : for a diffeomorphism, the
product of the derivatives along a period orbit is a
$\mathcal{C}^{1}$-conjugacy invariant. In particular, Anosov diffeomorphisms
or flows can easily be deformed to modify this conjugacy invariant, without
changing the topological structure (by structural stability). The absence of
periodic orbits is on the other hand possible (and actually prevalent) in
_entropy zero dynamics_ , which is therefore a natural setting to investigate
geometric rigidity.

A fundamental class of (entropy zero) systems in which geometric rigidity has
been shown to be prevalent are (minimal) _circle diffeomorphisms_ , a class of
dynamical systems that have played a central role in the development of the
theory since the work of Poincaré onwards. In addition to asking that there
are no-periodic orbits (which in the setting of circle diffeomorphisms is
equivalent to the assumption that the rotation number is _irrational_), to
prove that a diffeomorphism $T:M\to M$ is rigid one often needs to impose a
quantitative version of the absence of periodic orbits, for example asking
that there exists $c>0$ and $\alpha>0$ such that

(1) $d(x,T^{n}(x))\geq\frac{c}{n^{\alpha}},\qquad\mathrm{for\ all}\
n\in\mathbb{N},\ x\in M$

(where $d$ is a distance function on $M$). When $M$ is the _circle_
$\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}$, it is well known that a diffeomorphism
$T:\mathbb{S}^{1}\to\mathbb{S}^{1}$ satisfies (1) for a _full measure_ set of
_rotation numbers_ (actually with $\alpha>1$). In this setting, indeed, (1) is
equivalent to assuming that the rotation number $\rho=\rho(T)$ of $T$ is
_Diophantine_ 111We recall that one says that $\rho\in\mathbb{R}$ is
_Diophantine_ with Diophantine exponent $\tau>0$ iff there exists $c>0$ such
that for all $p,q\in\mathbb{Z}$, $q\neq 0$,
$\left|\rho-\frac{p}{q}\right|\geq\frac{c}{q^{2+\tau}}$. Since rotations are
homogeneous, multiplying by $q$ one gets (1) with $\alpha=1+\tau$. or, as
equivalent terminology, satisfies a _Diophantine Condition_. (For this reason,
an assumption like (1) is sometimes called a _Diophantine-type_ condition).

A celebrated result by Michael Herman [Herman], combined with later work by
Jean-Christophe Yoccoz [Yoccoz], then shows that circle diffeomorphisms which
satisfy a Diophantine Condition are _geometrically rigid_.

Let us briefly summarize some of the works and settings in which geometric
rigidity has been verified, that are perhaps surprinsigly few:

  1. (1)

In the above mentioned setting of circle diffeomorphisms, the first (local)
result in this direction of rigidity was obtained by Arnol’d in [Arnold] by
applying methods from KAM theory. The global theory was brought about by the
work of Herman [Herman] and completed by Yoccoz [Yoccoz]. It was later
revisited in terms of renormalization theory, see [KS:Her, KT:Her].

  2. (2)

If one allows to replace the ambient Riemannian manifold by a minimal
invariant closed set, certain smooth unimodal maps of the interval $[0,1]$ are
known to be geometrically rigid. These were first numerically discovered by
physicists Feigenbaum [Feigenbaum] and Coullet-Tresser [CoulletTresser] in the
late 1970s. A deep and rigorous theory, nowadays sometimes referred to as
_Sullivan-McMullen-Lyubich theory_ was established only later, in the
Nineties, through the introduction of complex methods into the picture, see
for instance Sullivan [Sullivan], McMullen [McMullen1, McMullen2], Lyubich
[Lyubich] and Avila-Lyubich [AvilaLyubich].

  3. (3)

In _one-dimensional dynamics_ , several other classes of rigid dynamical
systems were discovered and much studied, such as circle maps with a _critical
point_ (see e.g. the works by de Faria and de Melo [deFariadeMelo,
deFariadeMelo2] or by Yampolsky [Ya:att, Ya:hyp]), circle homeomorphisms which
are differentiable away from a point, known as circle maps with _breaks_ (see
for example the works [KhaninKhmelev, KhaninTeplinsky, KhaninKocicMazzeo] by
Khanin, Khmelev, Teplinsky, Kocic, Mazzeo, or Cunha and Smania [CS:ren,
CS:rig] for more breaks) or Lorenz maps (see Martens and Winckler, [Winckler,
MartensWinckler]).

  4. (4)

KAM theory establishes the local rigidity of Diophantine translations on the
torus $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ (see for instance
[Karaliolios] and references therein). However, no global rigidity result is
known in this context when $d\geq 2$.

In the setting $(4)$, geometric rigidity was proposed as a conjecture by
Raphaël Krikorian (see [Kh:sur]), who asked, when $M=\mathbb{T}^{d}$, with
$d\geq 2$, is a higher dimensional torus, whether given any
$\mathcal{C}^{\infty}$ diffeomorphism $T:M\to M$ which is topologically
conjugate to a translation of $\mathbb{T}^{d}$ and whose rotation vector
satisfies a Diophantine Condition,222A translation of
$\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ with rotation vector
$\rho=(\rho_{1},\dots,\rho_{d})$ is the map which sends
$x=(x_{1},\dots,x_{d})\in\mathbb{T}^{d}$ to
$x+\rho=(x_{1}+\rho_{1},\dots,x_{d}+\rho_{d})\in\mathbb{T}^{d}$. We say that
the vector $\rho$ satisfies a Diophantine condition with exponent $\tau>0$ if
there exists $c>0$ such that for every non-zero integer vector
$k=(k_{1},\dots,k_{d})\in\mathbb{Z}^{d}$, $|\langle
k,\rho\rangle|=|k_{1}\rho_{1}+\dots k_{d}\rho_{d}|\geq c/||k||^{\tau}$. then
the conjugacy is $\mathcal{C}^{1}$ and actually $\mathcal{C}^{\infty}$. A bold
generalization of this conjecture was suggested by Konstantin Khanin in his
ICM address [Kh:sur], namely that any minimal $T\in\mathrm{Diff}^{\infty}(M)$
where $M$ is a smooth, closed Riemannian manifold which satisfies a
Diophantine-type condition as in (1) is geometrically rigid. Little evidence
is available towards this conjecture and further obstructions other than
periodic orbits (related for example to the presence of invariant
distributions) may play a role in this greater generality.

###  1.2. Geometric rigidity in genus two

In this article we provide a new class of _geometrically rigid_ dynamical
systems, by proving a _global_ rigidity theorem for foliations on surfaces of
genus $2$ (Theorem A here below), or, more in general, for a broader class of
interval exchange maps (see Theorem B and the remarks afterwards). The result
for foliations is the following. We explain the meaning of some key words just
below (and refer the reader to § 6 for precise definitions of notions involved
in the statement).

######  Theorem A.

Let $S$ be a closed orientable surface of genus $2$ and $\mathcal{F}$ a
_minimal_ orientable foliation on $S$ of class $\mathcal{C}^{3}$, with non-
degenerate (Morse type) singularities. Under a full-measure333The space of
topological conjugacy classes of such foliations can be parametrised by
finitely many parameters, and here _full-measure_ means ’for almost every
parameter’ with respect to the Lebesgue measure. This notion of full measure
on minimal foliations is also related to the _Katok fundamental class_ , see §
6.2.3 for details. Furthermore, it corresponds to a full measure set of
(combinatorial) rotation numbers (see Definition 2.3.1) in the sense of
Definition 3.3.1. Diophantine-type condition, $\mathcal{F}$ is geometrically
rigid.

This result can be seen as a generalisation of Herman’s global rigidity
theorem for circle diffeomorphisms, reformulated in the language of minimal
foliations on the torus (as we explain at the end of the next section of this
introduction). We recall that in higher genus, foliations are necessarily
_singular_. _Minimality_ in this context means that all _bi-infinite_ leaves
are dense (see Definition 6.1.3). By _Morse-type_ singularities we mean that
the leaves of the foliation in a neighborhood of a singularity are level-sets
of a Morse function (i.e. a function with non-degenerate zeros). This is a
_generic_ (open and dense) condition. These assumptions imply in particular
that singular points of the foliation are _saddles_ 444Morse type
singularities are _simple saddles_ (with $4$-separatrices or _prongs_) and
_centers_ : centers are excluded since if there is a center, the foliation has
closed orbits in the neighbourhood of the center and thefore is not minimal. A
minimal foliation in genus is two can have either $2$ simple saddles (with $4$
prongs each), or one (degenerate) saddle with $6$ prongs. The assumption of
Morse singularities implies that we are in the first case, but is included in
Theorem A only to have a simpler statement: the case of one saddle with
$6$-prongs is also covered by Theorem $B$ stated below, see also § 6. and that
the _holonomy_ of the foliation around each saddle points is zero (see § 6 for
details). Finally, we define the measure class on minimal foliations with
respect of which the Diophantine-type condition (which is given by Definition
6.2.2) has _full measure_ in § 6.2.3.

Note that all the examples of geometrically rigid dynamical systems that we
listed above, with the exception of the _local_ rigidity results given by KAM
theory, are one-dimensional and combinatorially equivalent to either a
translation on a torus (in the case of circle diffeomorphisms, critical circle
maps, or circle maps with breaks), or an odometer (in the case of unimodal
maps and Lorenz maps). Our result is, to the best of our knowledge, the first
(global) rigidity result on surfaces of higher genus, which have a much
richer555This _richness_ can be formalized in various ways: flows on surfaces
are described by more _frequencies_ ; the combinatorial information in this
setting can be described _higher dimensional continued fraction algorithms_ ,
which produce cocycles in $SL(d,\mathbb{Z})$ with $d\geq 2$; the combinatorial
information can also be encoded in a Bratteli-Vershik diagram with $d\geq 2$
vertices, while odometers and rotations both correspond to Bratteli-Vershik
diagrams with $d=2$. Finally, in virtue of this higher dimensional nature, one
lacks in general Denjoy-Koksma inequality and a priori bounds, see a later
subsection of this introduction. combinatorial structure.

Before we state the other main results of this article (in § 1.5 and § 1.6),
we summarise the main results on diffeomorphisms of the circle and
(generalized) interval exchange transformations that motivated our work.

###  1.3. Diffeomorphisms of the circle and foliations on the torus.

Flows and foliations on surfaces have been a topic of interest since the work
of Poincaré, who singled out the analysis of flows on the torus as the
simplest toy-model to investigate the stability of the solar system. Poincaré
introduced the rotation number, which is an invariant which fully accounts for
the combinatorial structure of orbits of circle diffeomorphisms (and
equivalently flows on the torus).

The rigidity theory of circle diffeomorphisms was started by Denjoy in
[Denjoy]. Recall that two homeomorphisms $f,g:S^{1}\to S^{1}$ of the circle
$S^{1}$ are _topologically conjugate_ if there exists a homeomorphism
$h:S^{1}\to S^{1}$ (the _conjugacy_ map) such that $f\circ h=g\circ f$. Denjoy
in [Denjoy] proved that a sufficiently regular circle diffeomorphism $f$ with
irrational rotation number must be _topologically conjugate_ to the _rigid
rotation_ $R_{\rho}$ with the same rotation number $\rho$, given by $x\mapsto
R_{\rho}(x):=x+\rho$. The existence of a topological conjugacy implies in
particular that $f$ cannot have _wandering intervals_ , namely there does not
exist intervals $I\subset S^{1}$ such that the iterates $f^{n}(I)$,
$n\in\mathbb{Z}$ are all disjoint.

A landmark result is the _local rigidity_ theorem of Arnol’d [Arnold], who
successfully applied KAM theory to show that under a suitable Diophantine-type
condition on the rotation number $\rho$, sufficiently small analytic
deformations of $x\mapsto x+\rho$, whose rotation number is equal to $\rho$,
must be analytically conjugate to $x\mapsto x+\rho$. Arnol’d went on to
conjecture that such a statement should hold true without any assumption on
the closeness to rotations.

This _global rigidity_ conjecture was proved to be true in the (more general)
$\mathcal{C}^{\infty}$ setting666Herman in his thesis [Herman] considers not
just the not just $C^{\infty}$ regularity, but also $C^{r}$, for $r\geq 3$ as
well as the analytic settings. by Michael Herman [Herman] in a spectacular
treaty, whose legacy still lives on. A few years later, Jean-Christophe Yoccoz
[Yoccoz] succeeded in showing that Herman’s result indeed extends to all
_Diophantine numbers_ , thus providing the optimal arithmetic condition in the
smooth setting (and later on also the optimal condition in the analytic
setting, see [Yo:CIME]). Combining Herman’s [Herman] and Yoccoz’ [Yoccoz]
results, we have the following theorem:

######  Theorem. (Herman [Herman], Yoccoz [Yoccoz]).

If $\rho$ is a Diophantine number, then any
$T\in\mathrm{Diff}^{\infty}(S^{1})$ of rotation number $\rho$ is
$\mathcal{C}^{\infty}$-conjugate to $x\mapsto x+\rho$. In particular, smooth
circle diffeomorphisms of Diophantine rotation number are (geometrically)
rigid.

An equivalent geometric reformulation of the above theorem in the language of
foliations on surfaces is the following. Let $S$ be a torus, namely a _genus
one_ closed orientable surface. Then, given any _minimal_ (orientable)
foliation on $S$ which is topologically conjugate to a linear flow on the
torus with Diophantine rotation number777Let us recall that a _linear flow_ on
the torus $\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$ is the flow
$(\varphi_{t})_{t\in\mathbb{R}}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ given by
$\underline{x}\mapsto\underline{x}+t\underline{\alpha}$, where
$\underline{x},\underline{\alpha}\in\mathbb{T}^{2}$. This flow has rotation
number $\rho=\alpha_{2}/\alpha_{1}$. is smoothly conjugated (in the sense of
foliations888We recall that the regularity of conjugacy of foliations is
expressed in terms of the transverse structure; thus, this is equivalent to
the conjugacy of $f$ and $R_{\rho}$.) to the linear flow foliation. In
particular, the theorem implies that minimal foliations on _genus one_
surfaces under a full measure Diophantine-type condition are geometrically
rigid. It is this latter statement that is generalized by Theorem A to _genus
two_.

###  1.4. Flows in higher genus and interval exchange transformations.

The extent to which the theory of diffeomorphisms of surfaces and flows on the
torus generalises to flows on higher genus surfaces and their Poincaré maps is
a natural question. The objects which play the role of rigid rotations in this
context are standard interval exchange transformations (IETs for short),
orientation-preserving bijections of $[0,1]$ which are piecewise translations
(see Definition 2.1.2). These transformations naturally arise as Poincaré maps
of _linear flows_ on higher genus surfaces (namely translation flows on
translation surfaces, or, correspondingly, measured foliations999A translation
surface determines indeed a vertical (and a horizontal) measure foliation. The
leaves of the vertical linear flow are leaves of the measured foliation and
the other foliation determines a transverse invariant measure.), see § 2.1.6.
_Linear flows_ on (translation) surfaces in turn play the role of linear flows
on (flat) tori. The non-linear counterparts are _generalized interval exchange
transformations_ (or GIETs), which arise as Poincaré maps of minimal flows on
higher genus surfaces. Notice that in higher genus the presence of
singularities is unavoidable and the corresponding (orientable) foliations
have singularities (corresponding to _fixed points_ for the flow).

####  1.4.1. Renormalization and combinatorics.

To generalize Poincaré and Denjoy work, one needs first of all a combinatorial
invariant which extends the notion of rotation number. Such an invariant can
be produced by recording the combinatorial data of a renormalization process.
Renormalization operators in this context, similarly to the case of circle
diffeomorphisms, are obtained associating to a given a GIET $T:I\to I$ on
$I=[0,1]$, another GIET which is obtained by suitably choosing an subinterval
$I^{\prime}\subset I$ and considering the _induced map_ of $T$. The interval
$I^{\prime}$ is chosen so that the induced map is well defined and is again a
GIET $T^{\prime}$ of the same number of intervals. Correspondingly, at the
level of (minimal) flows (or foliations) on surfaces, this process corresponds
to taking a smaller Poincaré section. The image $\mathcal{R}(T)$ of $T$ under
the renormalization operator is then by definition the GIET acting on
$I=[0,1]$ obtained by _normalising_ , i. .e. conjugating by the affine
transformation which maps $I^{\prime}$ to $I$, so that the image is again a
GIET on $I$.

A classical algorithm to _renormalize_ standard IET is the _Rauzy-Veech
algorithm_ , also called _Rauzy-Veech induction_ (whose definition we recall
in § 2.5), first introduced by Rauzy [Ra:ech] and used starting from the
seminal papers by Veech [Ve:gau, Ve:inI] to study fine ergodic properties of
standard IETs, see e.g. [Zo:dev, AF:wea, Ch:dis]. The ergodic properties of
this renormalization dynamics in parameter space is by now well understood
(see e.g. [Zo:gau, Bu:dec, AGY, AB:exp], or [Yoc:B] for a brief survey).

Rauzy-Veech induction is well defined also on GIETs with no connections (as
defined in § 2.1.3) and can be used (as shown e.g. in [MMY, MarmiYoccoz], see
also the notes [Yoc:Clay]) to define a notion of _rotation number_ (see
Definition 2.3.1) and _irrationality_ for IETs and GIETs (see Definition
2.3.2). One can then show that two _irrational_ GIETs with the same rotation
number are _semi-conjugated_ , a result that we call Poincaré-Yoccoz theorem
(see Theorem 2.1).

####  1.4.2. Absence of a Denjoy theorem and wandering intervals.

One of the crucial differences between GIETs and circle diffeomorphisms,
though, is the absence of a _Denjoy Koksma inequality_ 101010We recall that
the _Denjoy-Koksma inequality_ is an ergodic-theoretic statement which gives
boundedness of Birkhoff sums of bounded variation observables at special
times: given $f:I\to I$ is a function of bounded variation on $I=[0,1]$ and
$R_{\rho}$ a rotation by $\rho$, if $p_{n}/q_{n}$ are the _convergents_ of
$\rho$ (given by $p_{n}/q_{n}=[a_{0},\dots,a_{n}]$ where
$[a_{0},\dots,a_{n},\dots]$ is the continued fraction expansion of $\rho$),
the Birkhoff sums $\sum_{k=0}^{q_{n}}f(x)$ at times $q_{n}$ are uniformely
bounded, independently on $n\in\mathbb{N}$ and $x\in I$. and _a priori
bounds_. This has far-reaching consequences, the most spectacular of which
being the absence of Denjoy theorem: there are smooth GIETs that are semi-
conjugate to a minimal IET for which the semi-conjugacy is not a conjugacy, in
other words they have wandering intervals. This phenomenon, first discovered
by Levitt but for a non-uniquely ergodic example, was later expored by
Camelier Gutierrez [CamelierGutierrez], Cobo [Cobo] and Bressaud, Hubert and
Mass [BressaudHubertMaass] for special (families of) uniquely ergodic examples
with periodic-type combinatorics. It is important to stress that this is is
not a low-regularity phenomena, nor it is related to special arithmetic
assumptions, as these examples exist for AIETs with _almost every_ rotation
number, as shown later in the work [MMY2] by Marmi, Moussa and Yoccoz, which
actually indicates that the existence of wandering interval is in some sense
_typical_ 111111See for example the statement of Proposition 5.3.1, which is
taken from [MMY]: wandering intervals are shown to exist for a full measure
set of rotation numbers as long as the log-slope vector of the AIET has a
typical projection on the Oseledets filtration (and, conjecturally, as long as
it projects on any positive Oseledets exponent)..

####  1.4.3. Cohomological equations and obstructions to linearisation.

A crucial step in the KAM approach developed by Arnol’d for circle
diffeomorphisms is to solve a linearised version of the conjugacy equation
$h\circ R_{\rho}=T\circ h$, which amounts to finding a (smooth) $f$ which
satisfy the functional equation $f\circ R_{\rho}-f=g$ for a given (smooth)
$g$. This equation, known as cohomological equation, is easily solved in the
smooth setting using Fourier analysis in the case where $R_{\rho}:=x\mapsto
x+\rho$ is a rotation satisfying a full measure arithmetic condition, under
the necessary (and in this setting the only) obstruction that $g$ has to have
zero-mean.

For a long time it has been unknown whether the cohomological equation could
be solved under suitable assumptions for IETs (or flows on surfaces such as
translation flows), until the pioneering work of Forni [Fo:ann] (see also
[Forni2]), who brought to light the existence of a finite number of
obstructions to solving it. The existence of obstructions to solve the
cohomological equation has been since then discovered to be a characteristic
phenomenon in _parabolic dynamics_ , see for example the works by Flaminio and
Forni on the cohomological equation for horocycle flows [FF] and [FF2] for
nilflows on nilmanfolds (which are other key examples of _parabolic_ flows, in
the sense that they present a subexponential form of sensitive dependence of
initial conditions, see for example the surveys [Ul:she, Fo:asy]). Forni’s
work is a breakthrough that paved the way for the development of a
linearisation theory in higher genus.

Another breakthrough was achieved by Marmi-Moussa-Yoccoz in their work [MMY3]
(and related works [MMY, MarmiYoccoz]). In [MMY], in particular, they reproved
Forni’s result on the cohomological equation using a renormalization approach
based on Rauzy-Veech induction, thus describing explicitely a full measure
Diophantine-type condition on the IET (a condition that, in analogy with
rotations, they called _Roth type_ , see [MMY] and also [MarmiYoccoz] for a
variation of this condition). Furthermore, the improved regularity in their
result could then be exploited in [MMY], combined with a generalization of
Herman’s _Schwarzian derivative trick_ , to prove a linearisation result,
showing that the high regularity ($\mathcal{C}^{r}$ for $r\geq 2$)-local
conjugacy classes of smooth IETs form a submanifold of the expected finite
codimension (the codimension being related to the number of obstructions to
solve the cohomological equation, see [MMY3]). An analogous result for the
$\mathcal{C}^{1}$-local conjugacy classes was suggested as a conjecture in
[MMY3]. Recently, the first author has proved it in a special case, namely for
the (measure zero) set of IETs which have (hyperbolic) periodic-type rotation
number (in the sense of Definition 2.3.3 below), which hence correspond to
periodic points of the renormalization operator.

####  1.4.4. Rigidity conjecture.

In [MMY3], Marmi, Moussa and Yoccoz formulate a number of fundamental open
questions and conjectures left open in the theory of linearisation of GIETs
(see the Open Problems section 1.2 in [MMY3]). One of them, stated as Problem
2 in [MMY3], is a geometric rigidity question/conjecture121212Immediately
before formulating it as a question, Marmi, Moussa and Yoccoz provide an
heuristic _rationale_ which explains why it should be true and, just after,
formulate a slight extension of what they now call ’one of the previous two
_conjectures_ ’.. They ask whether it is true that, for a full measure set of
standard IETs $T_{0}$, any GIET $T$ of class $\mathcal{C}^{4}$ to $T_{0}$ and
such that the value of a conjugacy invariant that they call _boundary_ (see
below and, for a definition, § 2.7.4) is zero131313More precisely, Problem $2$
in [MMY3] is first stated for GIETs which are a _simple deformation_ of
$T_{0}$ (i.e. a deformation which does not perturb $T_{0}$ in a neighbourhood
of the discontinuities and endpoints, see [MMY3]). Being a simple deformation
implies in particular that the boundary is the same than the boundary of
$T_{0}$ and the latter is indeed zero. Immediately after, they say that the
conjecture can be formulated in a slightly more general setting (not
restricted to simple deformations) using the boundary conjugacy invariant that
they introduce later., is actually also $\mathcal{C}^{1}$-conjugate to
$T_{0}$.

We prove in this paper that this conjecture is true in genus two (see Theorem
B below). We also show that the result in any genus can be reduced to a
statement on dynamical partitions of affine IETs, or equivalently, to problem
concerning Birkhoff sums of piecewise constant observables over standard IETs
(see the comments below, or § 5 and in particular Proposition 5.2.1).

###  1.5. Rigidity result for GIETs

We have already anticipated one of our rigidity results in the language of
foliations (Theorem A stated above). We will now formulate our main result in
the language of GIETs (Theorem B below).

We denote by $\mathcal{I}_{d}$ the space of standard irreducible interval
exchange transformations with $d$ branches (see § 2.1.3 for the definition of
irreducible). This space is a finite union of $d\\!-\\!1$ simplexes (see §
2.2) and thus carry a natural Lebesgue measure. Full-measure sets and full-
measure Diophantine-type conditions are defined using this measure. Associated
to a GIET $T$, there is an important $\mathcal{C}^{1}$-conjugacy invariant,
called the _boundary_ of $T$ and here denoted by $\mathcal{B}(T)$ (for the
definition of $\mathcal{B}$, which is based on Marmi-Moussa-Yoccoz _boundary_
operator from [MMY3, MarmiYoccoz], see Definition 2.7.1 and § A.1). Our main
rigidity result in the language of interval exchange maps is the following.

######  Theorem B (Rigidity of GIETs with $d=4$ or $d=5$).

There is a full measure141414Here the measure is the Lebesgue measure on the
parameter _standard_ IETs $\mathcal{I}_{d}$, i.e. a result holds for a full
measure set of IETs in $\mathcal{I}_{d}$, if it holds for all _irreducible_
combinatorial data and Lebesgue-almost every choice of _lenghts_ of the
continuity intervals. See § 2.5.1 for details. subset
$\mathcal{F}\subset\mathcal{I}_{4}\cup\mathcal{I}_{5}$ such that the following
holds. If $T_{0}\in\mathcal{F}$ and a $\mathcal{C}^{3}$-generalized interval
exchange map $T$, whose boundary $\mathcal{B}(T)$ vanishes, is topologically
conjugate to $T_{0}$, then the conjugacy between $T$ and $T_{0}$ is actually a
diffeomorphism of $[0,1]$ of class $\mathcal{C}^{1}$. In other words, almost
every standard irreducible IET with $4$ or $5$ continuity intervals is
geometrically rigid.

Thus, Theorem B proves the rigidity conjecture by Marmi-Moussa-Yoccoz [MMY3]
for irreducible IETs with $d=4$ or $d=5$ intervals (which correspond to
Poincaré sections of flows in genus two). The $d=5$ case implies Theorem $A$
(see § 6) and, more in general, for $d=4$, the analogous statement for minimal
orientable foliations in genus two with a degenerate saddle.  The set
$\mathcal{F}$ of _standard_ IETs, which has full measure with respect to the
_Lebesgue_ measure on $\mathcal{I}_{4}$ or $\mathcal{I}_{5}$ ( (see footnote
14 and § 2.5.1 for details) is described by a _Diophantine-type_ condition
that we call $(RDC)$. We comment on its nature below (see § 1 and Definition
3.3.4 for the precise condition).

We remark that a great part of the intermediate results which are proved to
deduce Theorem B (see e.g. Theorem C and Theorem E stated below), are proved
in greater generality, namely for any $d\geq 2$ (and hence for any genus in
the language of foliations in Theorem A). The result which is exploited in the
proof and reduces the validity of the rigidity conclusion to $d=4,5$ (and
respectively genus two foliations) is a result on existence of wandering
intervals for _affine IETs_ (and equivalently on the control of Birkhoff sums
of piecewise constant observables), which was proved by Marmi, Moussa and
Yoccoz in [MMY2] and known only under a technical condition on Lyapunov
exponents (which is automatically satisfied for $d=4,5$).

####  1.5.1. Regularity of the conjugacy

The reader familiar with the theory of circle diffeomorphisms will have
noticed that Theorem B only gives that the conjugating map is of class
$\mathcal{C}^{1}$. We believe that it should be possible to prove that the
regularity is indeed $\mathcal{C}^{1+\alpha}$ (i.e. that the derivative is
$\alpha$-Hölder) for some $0<\alpha<1$ but that indeed the conjugacy is not
typically $\mathcal{C}^{2}$. We stress that this is not due to a shortcoming
in our approach, rather this loss of regularity is an essential feature of the
problem, which corresponds to Forni’s and Marmi-Moussa-Yoccoz non-trivial
obstructions to solving the cohomological equation: Marmi-Moussa-Yoccoz have
indeed shown that asking for more regular conjugacy forces GIETs to live in
positive codimension submanifolds of the $\mathcal{C}^{1}$-conjugacy class;
the codimension is an exact reflection of the aforementioned obstruction. In
this a sense, GIETs are closer to essentially non-linear rigid dynamical
systems, such as unimodal maps and circle map with breaks or critical points,
for which the conjugacy is typically no more regular than $\mathcal{C}^{1}$
and actually $\mathcal{C}^{1+\alpha}$ for some $0<\alpha<1$ in general).

####  1.5.2. The boundary assumption

We remark that the _boundary condition_ (i.e. the assumption that
$\mathcal{B}(T)$ vanishes) is an essential assumption: two GIET that are
topologically conjugate but have different boundaries cannot be differentiably
conjugated, simply because the boundary is $\mathcal{C}^{1}$-conjugacy
invariant. We note, for the reader who is familiar with the one-dimensional
dynamics literature, that the assumption that $\mathcal{B}(T)$ is zero, in the
special case where $T$ is a circle maps with breaks, reduces to the classical
assumption that the _non-linearity_ $\eta_{T}$ (see § 2.4.1) has integral zero
and that the special pair $(T_{1},T_{2})$, where $T_{1},T_{2}$ are the two
branches of $T$, corresponds to a diffeomorphism without break points151515The
boundary $\mathcal{B}(T)=0$ is indeed a vector in $\mathcal{R}^{\kappa}$ (see
Definition 2.7.1 and § A.1), where $\kappa:=d-2g$ and $g$ is the genus of any
(minimal) surface flow which has $T$ as Poincaré section, see § 2.1.6. Asking
that the sum of the entries of $\mathcal{B}(T)$ is zero is equivalent to
asking that $\int_{0}^{1}\eta_{T}(x)\mathrm{d}x=0$ (see Lemma 4.2.3); when $T$
is a circle diffeomorphisms with break points (i.e. when the combinatorics of
the GIET is of _rotational type_ , the value of the entries of
$\mathcal{B}(T)$ are related to the values of the break points, and therefore
asking that $\mathcal{B}(T)$ is the _zero-vector_ means asking that there are
no breaks, see Remark 4.2.3.. The case where $\mathcal{B}(T)$ does not vanish
is equally interesting, and some comments are made in a subsequent paragraph.

Geometrically, when $T$ is the Poincaré map of a minimal foliation on a
surface, the boundary $\mathcal{B}(T)$ encodes the holonomy around the saddles
of the foliation (see §6). Thus, the assumption that $\mathcal{B}(T)$ is zero
is equivalent to the asking that the corresponding foliation has trivial
holomony around singularities (a condition that is automatic when the
singularities are level sets of Morse functions). It is using this remark that
Theorem A can be deduced from Theorem B (see § 6).

####  1.5.3. The $\mathcal{C}^{0}$-conjugacy class in parameter space

The main result of this article does not yet give a description of the
$\mathcal{C}^{0}$-conjugacy class of IETs in parameter space. It shows on the
other hand that the $\mathcal{C}^{0}$-conjugacy class of almost every IET
agrees with the $\mathcal{C}^{1}$-conjugacy class. Marmi, Moussa and Yoccoz
conjectured that, for almost every IET, the $\mathcal{C}^{1}$-conjugacy class
is a codimension $(d-1)+(g-1)$ submanifold of class $\mathcal{C}^{1}$ (see
[MMY3], Problem 1). As already mentioned earlier, a step towards this
conjecture has been taken in [Selim:loc], which shows that it is locally true
for (hyperbolic) periodic type IETs (see Definition 2.3.3). The main result of
the present article, combined with a complete proof of the conjecture, would
therefore automatically yield also a complete description of the
$\mathcal{C}^{0}$-conjugacy class of almost every genus two IET in parameter
space.

###  1.6. A dynamical renormalization dichotomy and the strategy of the proof.

The proof of the rigidity results (Theorem B and its geometric reformulation
in Theorem A) are based upon renormalization methods. We prove in particular
some results on the dynamics of renormalization and its consequences, which we
now state, which are valid for infinitely renormalizable GIETs with any number
$d\geq 2$ of intervals and we believe are of independent importance.

Let $\mathcal{X}^{r}_{d}$ denote the space of all GIETs of class
$\mathcal{C}^{r}$ on $d\geq 2$ intervals with an irreducible combinatorics
(see § 2.1.3 and § 2.2 for definitions). We consider, as renormalization
operator $\mathcal{R}:\mathcal{X}^{r}_{d}\to\mathcal{X}^{r}_{d}$, an
acceleration of Rauzy-Veech induction (which is in turn given by suitable,
linearly growing iterates of Zorich acceleration of Rauzy-Veech induction).
The general statement that we prove on the dynamics of this renormalization,
which is valid for any $d\geq 2$ (and hence, correspondingly, for Poincaré
sections of flows on surfaces of any genus) is, informally, the following
_dynamical dichotomy_ (we refer to Theorem 3.2 for a precise statement):

######  Theorem C (A priori bounds or affine shadowing dichotomy).

Let $T$ be a GIET in $\mathcal{X}^{r}_{d}$, $d\geq 2$, whose rotation number
satisfies a full-measure161616The measure on (combinatorial) rotation numbers
is induced here by the Lebesgue measure on standard IETs, see Definition 3.3.1
and § 3.3.1 for details. Diophantine-type condition which we call $(RDC)$.
Then there exists a bounded set $K$ such that one of the two possibility
holds.

  1. (1)

The iterated renormalizations $(\mathcal{R}^{n}(T))_{n\in\mathbb{N}}$ are
recurrent to the bounded set $K$.

  2. (2)

The iterated renormalizations $(\mathcal{R}^{n}(T))_{n\in\mathbb{N}}$ go to
infinity at an exponentially rate, and the orbit $(\mathcal{R}^{n}(T))$ is
well-approximated by that of an affine interval exchange map.

The notion of _full measure_ is defined § 3.3.1 (see in particular Definition
3.3.1). We comment below (in § 1.6.3) on the nature of the Diophantine-type
condition. Let us say here though that this full measure condition includes in
particular as a (measure zero) special case all _periodic-type_ combinatorics
(also known as _Fibonacci-type_ combinatorics in the one-dimensional
literature, see § 2.3.4 for definitions). A proof of Theorem C in this special
case is much easier and is included both for didactical purposes and for the
reader not interested in the technical subtlety of Rauzy-Veech induction (see
section § 3.2); the periodic-type case (see Definition 2.3.3) also yields a
stronger conclusion, namely the approximation in $(2)$ is up to a bounded
error (see Proposition 3.2.1).

In the first case, Case $(1)$, which we call the _recurrent_ case, one can
show that _a priori bounds_ on iterates of renormalization hold (see
Proposition 4.2.1). The heart of the work in Case $(2)$ is to construct a
vector $v=(v_{1},\dots,v_{d})\in\mathbb{R}^{d}$, that we call the _affine
shadow_. This vector is such that $e^{v_{i}}$, for $1\leq i\leq d$, are the
slopes of an affine IET whose orbit under renormalization gives the _leading
divergent behaviour_ of the orbit of $T$ (see Theorem 3.2 for a precise
statement). Thus, the quantities $v_{i}$, $1\leq i\leq d$ play the role of
_geometrical scaling invariants_ associated171717We remark though that the
shadow $v\in\mathbb{R}^{d}$ is not uniquely defined, but its _unstable
component_ (which leaves in a space of dimension $g$) is: two shadows
$v_{1},v_{2}$ of the same GIET $T$ differ by an element of the central stable
space $E_{cs}(T_{0})$ of the Oseledets filtration of the standard IET $T_{0}$
semi-conjugated to $T$, see § 3.3.2. to $T$. Theorem C is an instance of study
of an infinitely renormalizable dynamics, whose orbit diverges in parameter
space. Even in this context, describing the _way_ in which divergence occurr
proves to be helpful to control the dynamical behaviour of the system. An
interesting occurrence of this phenomenon in one-dimensional dynamics, similar
at least in spirit, has been recently analysed for certain Cherry flows, in
the work of Martens and Palmisano [MartensPalmisano].

####  1.6.1. Wandering intervals and a priori bounds

Another key step of the proof is to show that, if one can prove that the
affine IET that shadows $T$ given by $(2)$ has wandering intervals by showing
that the dynamical partitions associated to the AIET are _exponentially
distorted_ (a geometric notion that we define in § 5.2, see Definition 5.2.2),
then also the GIET $T$ has wandering intervals (see Proposition 5.2.1). Thus,
the problem of existence of wandering intervals for GIETs is reduced by our
work to a question181818We remark though that it is not sufficient for us to
simply show that the affine shadow has wandering intervals, but we need to
show that this happens in a _special way_ , namely one needs to show the
Birkhoff sums estimates proved by Marmi-Moussa and Yoccoz in [MMY2], as stated
in Proposition 5.3.1, or, equivalently, that dynamical partitions are
_exponentially distorted_ in the sense of Definition 5.2.2. It is possible
that this is indeed the only way in which a wandering interval can appear in
an affine interval exchange transformation, but this may be difficult to
prove. concerning _affine_ interval exchange transformations, or more
precisely Birkhoff sums of piecewise constant functions over standard IETs.

Since Marmi, Moussa and Yoccoz have shown that a large class of AIETs have
exponentially distorted towers and hence wandering intervals (see § 5.3.2 and
in particular Proposition 5.3.1), it follows that all GIETs which are shadowed
by AIETs in this class (which includes typical AIETs for any $d\geq 2$, see
Proposition 5.3.1) have wandering intervals. When the number of exchanged
intervals is $d=4$ or $d=5$ (i.e. when the GIET is a Poincaré section of a
minimal flow on a genus two surface), the result by Marmi, Moussa and Yoccoz
[MMY2] include in particular all AIETs with divergent shadow191919More
precisely, the condition $v\in E_{2}(T)\backslash E_{3}(T)$ in Proposition
5.3.1 is automatically satisfied, see the proof of Theorem 5.4.. Thus, in this
case, assuming that $T$ is minimal (an assumption which rules out the presence
of wandering intervals) forces $T$ to be recurrent, i.e. Case $1$ of the
dynamical dichotomy given by Theorem C to hold. Thus, we can deduce in this
case a result on _a priori bounds_ :

######  Theorem D (A priori bounds in genus two).

If $T$ is a _minimal_ GIET in $\mathcal{X}^{r}_{d}$ with $d=4$ or $d=5$ whose
rotation number satisfies the full-measure condition $(RDC)$, then the
acceleration $\mathcal{R}$ of the Rauzy-Veech renormalization satisfies _a
priori bounds_ , namely there exists a constant $K>0$ such that the iterates
$\mathcal{R}^{m}(T)$ of $T$ under renormalization satisfy

$K^{-1}\leq||\mathrm{D}\mathcal{R}^{m}(T)||_{\infty}\leq K,\qquad\textrm{for\
all}\ m\in\mathbb{N},$

where $\|\cdot\|_{\infty}$ denotes the sup norm on $I=[0,1]$.

We refer to § 4.2.5 (in particular Propostion 4.2.1) for a more precise
formulation.

Generalizing the aforementioned result by Marmi, Moussa and Yoccoz [MMY2] (in
particular Proposition 5.3.1) to cover all divergent shadows202020More
precisely, one needs to show that Proposition 5.3.1 holds for any shadow which
has a projection on a positive Lyapunov exponent, which is not necessarily the
second as in the case when one assumes that $v\in E_{2}(T)\backslash
E_{3}(T)$. for a full measure set of AIETs with any $d\geq 2$ will remove the
restriction that $d=4,5$ from the statement of Theorem D. Notice on the other
hand that _no boundary condition_ appears in Theorem D, nor in Theorem C. The
assumption that the boundary $\mathcal{B}(T)$ is zero is indeed only required
when we proceed to prove a conjugacy regularity result.

####  1.6.2. Boundary obstructions and convergence of renormalization

The next conceptual step of our proof is to show that, when one is in Case
$(1)$, namely the recurrent case of the dynamical dichotomy Theorem C (for
example because one has ruled out case $(2)$ by showing that it would imply
the presence of wandering intervals and hence non-minimality), one can prove
results on exponential convergence of renormalization. More precisely, we show
the following result, which holds for any $d\geq 2$:

######  Theorem E (Exponential convergence of renormalization).

Let $T$ be a GIET in $\mathcal{X}^{r}_{d}$ whose rotation number satisfies the
full-measure condition $(RDC)$. Assume that $T$ satisfy the conclusion $(1)$
of Theorem C and that the boundary $\mathcal{B}(T)$ is zero. Then the orbit
$(\mathcal{R}^{m}(T))_{m\in\mathbb{N}}$ of $T$ under renormalization converges
exponentially fast, in the $\mathcal{C}^{1}$ distance, to the subspace
$\mathcal{I}_{d}$ of (standard) IETs.

The precise formulation of the theorem and the definition of $\mathcal{C}^{1}$
distance are given in § 4 (see in particular Theorem 4.1 and § 4.2.1). In this
case, we can then conclude, using classical arguments, that $T$ is
$\mathcal{C}^{1}$-conjugated to a standard IET with the same rotation number
(as shown in § 5.1).

We consider the proof of Theorem E to be a streamlined presentation and
generalization to GIETs of the now classical theory of Herman [Herman] for
circle diffeomorphisms. Some of these steps are well known in the literature
on circle diffeomorphisms with singularities or are folklore, other require
some variations of the arguments which are specially required to deal with the
increased complexity of GIETs.

We first show that, under the assumptions of Theorem E, the dynamical
partitions associated to the GIET (whose definition is given in § 2.3.7)
converge exponentially fast to the trivial partition into points (i.e. their
_mesh_ , or the size of the largest interval, decay exponentially). This can
be seen also as a generalization of the arguments by Cunha and Smania in
[CS:ren] for a measure zero class of circle diffeos with break points (those
which correspond to _bounded-type_ , rotational GIETs) to almost every
rotation number and, more in general, to almost every GIET which satisfies a
priori bounds thanks to the recurrence given by the conclusion of Case $(1)$
of Theorem C.

Exponential decay of the mesh can be used, as in the classical theory of
circle diffeomorphisms and its extensions to diffeos with singularities, to
show that iterates of renormalization converge to the space of Moebius IETs
(GIETs whose branches are Moebius functions, see 2.1.4). These first two steps
do not require the assumption that $\mathcal{B}(T)$ is zero.

The boundary assumption becomes essential to proceed further. Indeed,
requiring that $\mathcal{B}(T)$ is zero restricts us to a positive
codimension, renormalization invariant subset of the total space of GIETs
which contains standard IETs. We call this the linear regime, in contrast to
the non-linear regime (see § 4 for the precise definitions). In the linear
regime we show indeed that one is attracted to the space of _affine_ IETs
first, and actually, in a second step, to the space of standard IETs
$\mathcal{I}_{d}$.

The distinction between _linear_ (boundary zero) and _non-linear_ (boundary
non-zero) regime is a generalisation of the difference between standard circle
diffeormorphisms and circle maps with breaks, whose renormalization theory
extends in a non-trivial way that of circle diffeomorphisms. We believe that
the study of GIETs and renormalization in the non-linear regime is also very
interesting and, to the best of the authors knowledge, very little is known in
this regime. The renormalization dynamics has in this case a natural attractor
which is the set of Moebius IETs, but the dynamics of the renormalization
operator in that case is much more intricate to analyse.

####  1.6.3. The Diophantine-type condition

Finally, we comment on the Diophantine-type condition appearing in Theorem C
(which is also the full measure condition implicitely underlying Theorem A and
Theorem B). The full measure condition, that we call _Regular Diophantine
Condition_ , or $(RDC)$ for short, is formulated in terms of the Zorich (also
known as Zorich-Kontsevich) cocyle over the induction. At each step of
renormalization one can associate a matrix $Z(n)\in\mathrm{SL}(d,\mathbb{Z})$.
These matrices can be considered as a multi-dimensional generalisations of the
coefficients $a_{n}$ appearing in the continued fraction expansion of a
rotation number. The Diophantine-type condition has two aspects (as many
Diophantine-type conditions introduced for IETs and GIETs, see e.g. [MMY,
MarmiYoccoz]):

  1. (1)

A _growth_ condition, which straightforwardly generalises arithmetic,
Diophantine-type conditions in the genus $1$; one asks that the matrices
$Z(n)$ do not grow too fast (subexponentially with $n$ in our case).

  2. (2)

A _Oseledets_ aspect, which is specific to the higher genus case: we demand
that the product of the matrices $Z(n)\cdots Z(2)Z(1)$ is generic with respect
to Oseledets theorem in a _quantitative_ way.

  3. (3)

A _quantitative recurrence_ aspect, where certain series depending on whole
history of the Zorich-Kontsevich cocycle are required to be uniformely bounded
along a subsequence of renormalization iterates.

Our condition is in part reminiscent of the Roth type condition and restricted
Roth type conditions introduced by Marmi-Moussa-Yoccoz (see [MMY] and
[MarmiYoccoz] respectively) and fairly similar in spirit (Roth type conditions
also have a growth condition, usually denoted condition $(a)$, as well as
further conditions, like condition (b) and (c) in the standard Roth type
condition, which can be inferred from Oseledets genericity), but it is
significantly more restrictive.

First of all, we require a _quantitative_ version of the conclusion of
Oseledets theorem, in which the convergence is made _effective_ (see § 7.2).
For technical reasons, we work with the natural extension (by choosing an
arbitrary past for the rotation number) and require the existence of an
effective Oseledets generic extension. When the (extended) rotation number is
generic with respect to this effective version of Oseledets, one can show that
certain series, which depend on the whole matrices of the cocycle (explicitely
given by the _forward_ series _backward_ series $(F)$ and $(B)$ in the
Definition 3.3.4 of the $(RDC)$ condition), are _finite_. The above mentioned
_recurrence_ amounts to the request that infinitely often, along a linearly
growing subsequnce of times of the Zorich acceleration, these series are
_uniformely bounded_. Conditions of similar (albeit simpler) nature on
standard IETs were used by the second author in her work [Ul:abs] on absence
of mixing for special flows over IETs and appear as well in recent results in
her joint work with K. Fraczek [FU:gro] on deviations of Birkhoff averages for
locally Hamiltonian flows.

Examples of arithmetic conditions on classical rotation numbers which do not
depend only on the asymptotic behaviour of the continued fraction entries (as
_Diophantine_ or _Roth-type_ conditions) but instead depend on the _whole_
record of the continued fraction entries are for example the _Brjuno_
-condition (see e.g. [Yo:CIME]) or the _Perez-Marco_ condition [PM:sur]. In
the theory of circle diffeos, conditions which require recurrence to a set of
rotation numbers with this type of control on the whole history seem to appear
in global rigidity results, see for example the Condition $(H)$ defined by
Yoccoz (see [Yo:CIME]).

While our condition is full measure (in the sense of Definition 3.3.1), it is
likely not optimal. It would be interesting, but probably very difficult, to
describe the optimal Diophantine-type condition for a GIET to satisfy the
dynamical dichotomy in Theorem C.

###  1.7. Organization of the paper and reading guide.

In the _background_ Section 2 we give basic definitions, in particular
defining GIETs (as well as IETs, affine IETs and Moebius IETs), Rauzy-Veech
induction for GIETs, infinitely renormalizable GIETs, irrationality and
rotation numbers. We also summarize a number of classical tools and results
which are used in the rest of the paper. These include both tools from the
classical theory of circle diffeos and one dimensional dynamics (such as
distorsion, distorsion bounds, non-linearity and Schwarzian derivative) as
well as renormalization tools for IETs and GIETs related to Rauzy-Veech
induction, such as Zorich acceleration, invariant measures for the dynamics on
parameter spaces, dynamical partitions and Rohlin towers produced by Rauzy-
Veech induction, special Birkhoff sums and decomposition of special Birkhoff
sums. This section does not contain any new result. The reader familiar with
one or both these backgrounds can skip this section or read it only quickly as
a notational reference.

Section $2$ contains the precise formulation and the proof of the _dynamical
dichotomy_ stated informally in this introduction as Theorem C. In § 3.2 we
first state and prove a (stronger) dynamical dichotomy in the special case of
_bounded type_ rotation numbers (or Fibonacci combinatorics), defined in §
2.3.4. This proof can be skipped by the reader interested in the full measure
result. We decided to present it first, even though it lenghten the paper,
since it can be accessible to the reader that is not familiar with Rauzy-Veech
induction and already present all the key difficulties and ideas of the
general proof. The general case requires the definition of full measure set of
GIETs and rotation numbers and the definition of the Regular Diophantine
Condition $(RDC)$, which are given in § 3.4 In § 3.5.1 we can then give the
precise formulation of Theorem C in the general case, which is Theorem 3.2.
The rest of the section is devoted to the proof. An outline of the main steps
of the proof are given in § 4.1.

The main result of Section $3$ is Theorem E on _exponential convergence of
renormalization_ in the recurrent case. The proof takes the whole section and
is split in several steps, such as a priori bounds (§ 4.2.5) exponential decay
of the dynamical partitions mesh in § 4.3 and convergence first to Moebius
IETs in § 4.4.1, then to AIETs (see § 4.5.1) and finally to IETs in § 4.6.

In Section $4$, we prove the _rigidity result for GIETs_ , namely Theorem B of
this introduction. On one hand we prove that, when one has exponential
convergence of renormalization and the $(RDC)$ Diophantine-type condition, one
can deduce that the conjugacy is $\mathcal{C}^{1}$. This is done in § 5.1. On
the other hand, in § 5.2, we deduce the existence of wandering intervals for a
GIET from _exponential distorsion_ of the dynamical partitions of the affine
shadow, see Proposition 5.2.1, stated in in § 5.2.3 and proved in § 5.3.3.
Combined with the results on wandering intervals proved by Marmi-Moussa and
Yoccoz (recalled in § 5.3.1), this allows us to finish the proof of the
rigidity result for GIETs as well as Theorem D on a priori bounds in genus two
(in § 5.5).

In Section $5$ we prove Theorem A on foliations on surfaces of genus two. We
first define foliations, their regularity and their holomomies. We then deduce
Theorem A for foliations in genus two from Theorem B on GIETs with $d=4,5$.

In the Appendix, we include for convenience of the reader the proof of the
(extension of) some classical results, such as the distorsion bounds for
GIETs, the comparision between some of the distances used in Section 4, as
well as some results from [Selim:loc], in particular on Lipschitz regularity
of the renormalization operator, which are used in Section 4.

##  2\. Background material

###  2.1. Interval exchange transformations

The piecewise differentiable maps which arise as Poincaré maps of a smooth,
orientable foliation on a transversal interval are know as _generalized
interval exchange transformations_ 212121The name _generalized_ interval
exchange maps is used since they generalize _interval exchange
transformations_ (see Definition 2.1.2 below), which appear as Poincaré
sections of _measured foliations_ on transverse intervals, in suitably chosen
coordinates..

####  2.1.1. Generalized interval exchange transformations

Let us start by recalling the definition of generalized interval exchange
transformations, or, for short, GIETs.

######  Definition 2.1.1 (GIETs).

Let $d\geq 2$ be an integer and $r$ a positive real number. A
$\mathcal{C}^{r}$-generalized interval exchange transformation (GIET) of $d$
intervals, or for short a $d$-GIET of class $r$, is a map $T$ from the
interval $[0,1]$ to itself such that:

  * (i)

there are two partitions (up to finitely many points) of
$[0,1]=\bigcup_{i=1}^{d}{I^{t}_{i}}=\bigcup_{i=1}^{d}{I^{b}_{i}}$ of $[0,1]$
into $d$ open disjoint subintervals, called the _top_ and _bottom_ partition;
the subintervals are denoted respectively $I_{i}^{t}$, for $1\leq i\leq d$,
and $I_{i}^{b}$, for $1\leq i\leq d$;

  * (ii)

for each $1\leq i\leq d$, $T$ restricted to $I_{i}^{t}$ is an orientation
preserving diffeomorphism onto $I_{i}^{b}$ of class $\mathcal{C}^{r}$;

  * (iii)

$T$ extends to the closure of $I_{i}^{t}$ to a
$\mathcal{C}^{r}$-diffeomorphism onto the closure of $I_{i}^{b}=T(I_{i}^{t})$.

We will call the restriction $T_{i}:=T|I^{t}_{i}$ of $T$ onto $I^{t}_{i}$, for
$1\leq i\leq d$, a _branch_ of $T$. We think of $j\in\\{1,\dots,d\\}$ as the
_label_ of the intervals $I_{j}^{t}$ and $I_{j}^{b}=T(I_{j}^{t})$ and denote
by $\mathcal{A}:=\\{1,\dots,d\\}$ be the alphabet consisting of labels. Notice
that $T$ is by construction invertible and that the inverse $T^{-1}$ is also a
$\mathcal{C}^{r}$-GIET, for which the top and bottom partition are reversed.

####  2.1.2. Standard, affine and Moebius IETs

Special cases of generalized interval exchange transformations include
standard interval exchange transformations (IETs), affine interval exchange
transformations (AIETs) and Moebius interval exchange transformations (MIETs):

######  Definition 2.1.2 (IETs).

A GIET $T$ is an (_standard_) _interval exchange transformation_ or a $IET$ if
$|I^{t}_{i}|=|I^{b}_{t}|$ for every $1\leq i\leq d$ and the branches $T_{i}$
of the map $T$, for every $1\leq i\leq d$, are assumed to be _translations_ ,
i.e. of the form $x\to x+\delta_{i}$ for some $\delta_{i}\in\mathbb{R}$.

######  Definition 2.1.3 (AIETs).

A GIET $T$ is an _affine interval exchange transformation_ or an $AIET$ the
branches $T_{i}$ of the map $T$, for $1\leq i\leq d$, affine map, i.e. of the
form $x\mapsto a_{i}x+b_{i}$ for some $a_{i},b_{i}\in\mathbb{R}$.

######  Definition 2.1.4 (MIETs).

A _Moebius IET_ $T$ is a generalized interval exchange transformation $T$ such
that the branches $T_{i}$, for $1\leq i\leq d$, are restrictions of _Moebius
maps_ , i.e. maps of the form

$x\mapsto m(x):=\frac{ax+b}{cx+d},\qquad\text{where}\quad ad-bc>0.$

Interval exchange transformations appear naturally as Poincaré first return
maps of orientable foliations on a surface on transversal segments. The
discontinuities arise indeed from points on the interval which hit a
singularity of the foliation (or an endpoint of the transversal interval) and
therefore do not return to the transversal, while the intervals $I_{j}^{t}$
are continuity intervals of the Poincaré map. The smoothness $r$ of the
branches depends on the regularity of the foliation. When the foliation is a
measured foliation, one can choose coordinates so that the Poincaré map is a
standard IET, while affine IETs are Poincaré maps of _dilation surfaces_ (see
for example the survey [Selim:survey]).

####  2.1.3. Combinatorial data

To encode the _order_ of the intervals (from left to right) at the top and
bottom partition of a GIET, we adopt the convention (which became standard
after its introduction in [MMY]) of using _two permutations_ , $\pi_{t}$ and
$\pi_{b}$ of $\\{1,\dots,d\\})$: $\pi_{t}$ (resp. $\pi_{b}$) describes the
order of the intervals in the top (resp. bottom) partition, so that, in order
from left to right, they are

$I^{t}_{\pi_{t}(1)},I^{t}_{\pi_{t}(2)},\dots,I^{t}_{\pi_{t}(d)}\qquad\text{(resp.\leavevmode\nobreak\
}I^{b}_{\pi_{b}(1)},I^{b}_{\pi_{b}(2)},\dots,I^{b}_{\pi_{b}(d)}).$

We call the datum $\pi:=(\pi_{t},\pi_{b})$ of these pairs of permutation the
_combinatorial datum_ of $T$, or simply the _permutation_ of $T$ (by abusing
the terminology, even though it is a actually a _pair_ of permutations). The
composition $\pi_{b}^{-1}\circ\pi_{t}$, also called _monodromy_ , is a
permutation (in the classical sense) which encodes how $T$ rearranges the
partition intervals. The choice of keeping track of a _pair_ of permutations
(instead than only the monodromy, which was used classically for IETs (see
[Ve:gau] or [Ra:ech]), allows to keep track of labels of intervals and plays a
crucial role in the definition of irrationality of rotation numbers of GIETs
(see Definition 2.3.2).

We will assume that the combinatorial datum is _irreducible_ , i.e. for every
$1\leq k<d$ we have

${\pi_{t}\\{1,\dots,k\\}\neq\pi_{b}\\{1,\dots,k\\}}$

(this guarantees in particular that the GIETs cannot be reduced to a GIET of a
smaller number of exchanged intervals). We will denote by
$\mathfrak{S}_{d}^{0}$ the set of irreducible combinatorial data
$\pi=(\pi_{t},\pi_{b})$ with $d$ symbols.

####  2.1.4. Singularities

We denote by $u_{i}^{t}$, for $0\leq i\leq d$ the endpoints of the top
partition intervals and, respectively by $u_{i}^{b}$, $0\leq i\leq d$, the
endpoints of the bottom partition, in their natural order, so that

$\displaystyle 0:=u^{t}_{0}<u^{t}_{1}<\cdots<u^{t}_{{d-1}}<u^{t}_{{d}}:=1;$
$\displaystyle 0:=u^{t}_{0}<u^{t_{1}}<\cdots<u^{t}_{{d-1}}<u^{t}_{{d}}:=1.$

Then, with the chosen conventions, we have

$I^{t}_{\pi_{t}(j)}=(u^{t}_{j-1},u^{t}_{j}),\qquad
I^{b}_{\pi_{b}(j)}=(u^{t}_{j-1},u^{t}_{j}),\qquad\text{for}\ 1\leq j\leq d.$

We will denote by $|J|$ the length (with respect to the Lebesgue measure) of
an interval $J\subset I$, so $|I^{t}_{\pi_{t}(j)}|=u_{j}^{t}-u_{j-1}^{t}$ and
$|I^{b}_{\pi_{b}(j)}|=u_{j}^{b}-u_{j-1}^{b}$. The points
$u^{t}_{1},\dots,u^{t}_{d-1}$ separating the top intervals are called the
_singularities_ of $T$. The points $u^{b}_{1},\dots,u^{b}_{d-1}$ are the
singularities of $T^{-1}$.

####  2.1.5. Connections

A _connection_ is a triple $(u^{t}_{j},u^{b}_{i},m)$ where $m$ is a positive
integer such that $T^{m}(u^{b}_{j})=v^{t}_{i}$. Thus, a connection encodes a
finite orbit whose starting point and end point belong to the set of endpoints
points $\\{u_{0}^{t},u_{1}^{t},\cdots,u_{d}^{t}\\}$. We say that $T$ has _no
connections_ if no such triple exists. This condition is also called _infinite
distinct orbit condition_ or _Keane condition_ for standard IETs. Keane indeed
proved that a standard IET with irreducible $\pi$ and no connections is
minimal. When $T$ is the Poincaré map of a transveral a flow along the leaves
of a foliation on $S$, connections correspond to _saddle connections_ on $S$,
i.e. trajectories of the flow which connect two singularities. Thus, if the
flow along the leaves of the foliation has no saddle connections, any GIET
obtained as Poincaré map has no connections.

####  2.1.6. GIETs, surfaces and foliations

A generalized interval exchange map $T$ can be _suspended_ [Ma:int, Yoc:Clay]
(see also Appendix A.1) to an orientable (singular) foliation
$\mathcal{F}=\mathcal{F}(T)$ on a closed oriented surface $S=S(T)$ such that
the singular points of $\mathcal{F}$ are (possibly degenerate) saddles (with
an even number of prongs); we can make this construction so that all the
discontinuity points of $T$ belong to singular leaves of the foliation (see
Appendix A.1). Furthermore, if the permutation is _irreducible_ , both
endpoints also belong to a singular leaf. If $T$ is minimal, the associated
foliation is minimal in the sense that all regular leaves are dense (see
Definition 6.1.3 in § 6). We denote by $\mathrm{Sing}=\mathrm{Sing}(T)\subset
S$ or $\mathrm{Sing}(S)\subset S$ the the set of saddle points of
$\mathcal{F}$. If $g$ is the genus of $S$ and $\kappa=|\mathrm{Sing}(S)|$ we
have the following equality

$d=2g+\kappa-1.$

Notice that both the genus $g$ and the number $\kappa$ of singularities can be
recovered purely combinatorially from the knowledege of the combinatorial
datum $\pi$, see [MarmiYoccoz, Yoc:Clay] or [Vi:IET] or Appendix A.1 for
details. Conversely, $T$ can be recovered from $\mathcal{F}$ by considering a
first-return map on a suitably chosen transverse arc $J$ in $S$ joining two
singularities in $\mathrm{Sing}(S)$, which we can identify with the interval
$[0,1]$ (see also Lemma 6.1.1). We develop this foliation point of view
further in Section 6. Notice that choosing $J$ with endpoints at
singularities, or on singular leaves, guarantees that the number of exchanged
intervals of $T$ is as small as possible and equal to $d$.

Let $\mathcal{F}$ be a foliation on $S$ which suspends $T$. We can associate
to each discontinuity of $T$ a singularity as follows. Let us say that
$\mathcal{F}$ is a _standard suspension_ if both endpoints222222Usually, in
the IETs and translation surfaces literature, one often uses as suspensions
the _zippered rectangles_ introduced by Veech [Ve:gau] (see e.g. [Yoc:Clay] or
[Vi:IET]) or the polygonal suspensions by Masur [Ma:int]. In zippered
rectangles, one endpoint is always at a singularity, while the other is
usually not and belong to a separatrix, i.e. a singular leave, that is either
ingoing or outgoing. From the foliation point of view, all suspensions whose
end points are on singular leaves are equivalent and one can simply slide the
singularity and assume that it is at an endpoint. This is convenient since it
makes it easier to identify singularities of the foliation with endpoints and
discontinuities of the GIETs. Standard (translation surfaces) suspensions are
for example explicitely defined in [MarmiYoccoz]; the construction is included
in Appendix A.1. suspension of $I$ are singular points of $\mathcal{F}$. In
this case, all the singularities of $T$ are obtained by pulling-back a
singular leaf232323If $\mathcal{F}$ is not standard, there can be
singularities which are created by an endpoint of the section, namely which
belong to the leave passing through an endpoint.. Then we can define a map $s$
from the set $\\{u_{0}^{t},\dots,u_{d}^{t}\\}$ of singularities and endpoints
of $T$ to the singularity set $\mathrm{Sing}(S)$ simply associating to the
endpoints $u_{0}^{t}$ and $u_{d}^{t}$ the corresponding singularity (i.e. the
endpoint of the section in $S$) and to all other $u_{i}^{t}$, $1\leq i<d$, the
singularity, that we will denote $s(u_{i}^{t})$, that is reached when
following the oriented leaf emanating from $u_{i}^{t}$.

###  2.2. Parameter (sub)spaces

We define, for a given differentiability class $r\in\mathbb{R}_{+}$ and number
of intervals $d\geq 2$ the space $\mathcal{X}^{r}$ of generalized interval
exchange transformations of class $\mathcal{C}^{r}$ with $d$ intervals, namely

$\mathcal{X}^{r}:=\bigcup_{\pi\in\mathfrak{S}_{d}^{0}}{\mathcal{X}_{\pi}^{r}},\qquad\text{where}\quad\mathcal{X}_{\pi}^{r}:=\\{T,\quad
T\ \text{d-GIET of class}\ \mathcal{C}^{r}\ \text{with associated permation}\
\pi\\}.$

When there is no ambiguity on the differentiability class $r$, we denote
$\mathcal{X}^{r}_{\pi}$ and $\mathcal{X}^{r}$ simply $\mathcal{X}_{\pi}$ and
$\mathcal{X}$ respectively.

The space of _(standard)_ interval exchange transformations (respectively,
_affine_ interval exchange transformations or _Moebius_ interval exchange
transformations) with combinatorics $\pi$ will be denoted by
$\mathcal{I}_{\pi}$ (respectively, $\mathcal{A}_{\pi}$ or $\mathcal{M}_{\pi}$)
and are subspaces of $\mathcal{X}^{r}$ for every $r\geq 0$. Similarly, for any
$d\geq 2$, let us set

$\mathcal{I}_{d}:=\bigcup_{\pi\in\mathfrak{S}_{d}}{\mathcal{I}_{\pi}},\qquad\mathcal{A}_{d}:=\bigcup_{\pi\in\mathfrak{S}_{d}}{\mathcal{X}_{\pi}^{r}},\qquad\mathcal{M}_{d}:=\bigcup_{\pi\in\mathfrak{S}_{d}}{\mathcal{M}_{\pi}}.$

Clearly, for every $r>0$, we have the inclusions

$\mathcal{I}_{d}\subset\mathcal{A}_{d}\subset\mathcal{M}_{d}\subset\mathcal{X}^{r}_{d}.$

####  2.2.1. Parameter spaces of IETs

If $T\in\mathcal{I}_{d}$ is a (standard) IET, $T$ is completely determined
from the combinatorial datum $\pi$ and the lengths of the top intervals
$I^{t}_{j}$, $1\leq j\leq d$. Denote by $\lambda_{1},\cdots,\lambda_{d}$ the
lengths of its continuity intervals, so that
$\lambda_{j}:=|I^{t}_{j}|=v^{t}_{j}-v^{t}_{j-1}$. Because the top intervals
form a partition of $[0,1]$, the lengths must satisfy the following equation:

(2) $\lambda_{1}+\cdots+\lambda_{d}=1.$

We will denote by $\Delta_{d-1}$ (for $d-1$ dimensional simplex) the set of
vectors in $\mathbb{R}^{d}_{+}$ which satisfy (2). We denote by $\lambda(T)$
and call _lengths vector_ the vector whose components are lengths of top
intervals, namely

$\lambda(T):=(\lambda_{1},\dots,\lambda_{d})=(|I^{t}_{1}|,\dots,|I^{t}_{d}|)\in\Delta_{d-1}.$

Thus, the subspace $\mathcal{I}_{d}$ of $d-$IETs is parametrized by
$\Delta_{d-1}\times\mathfrak{S}_{d}$ and $\mathcal{I}_{d}$ is a (finite union
of) submanifold(s) of $\mathbb{R}^{d}$ of dimension $d-1$.

####  2.2.2. Parameters of AIETs

Let $T$ be an AIET with combinatorial datum $\pi$. Let
$\lambda\in\Delta_{d-1}$ be as before the vector of lengths of top intervals.
If we denote by $\rho_{1},\cdots,\rho_{d}$ the derivatives of $T$ on intervals
of respective lengths $\lambda_{1},\cdots,\lambda_{d}$, we have that for each
$1\leq j\leq d$, the length $|I_{j}^{b}|$ of the bottom interval
$I_{j}^{b}=T(I_{j}^{t})$ is $\rho_{j}\lambda_{j}$. Therefore, since also the
bottom intervals form a partition, the lengths must also satisfy

$\rho_{1}\lambda_{1}+\cdots+\rho_{d}\lambda_{d}=1.$

This equation, together with (2) and the further restrictions that $\forall
i,\lambda_{i}>0$ identify $\mathcal{A}_{\pi}$ to a submanifold of
$\mathbb{R}^{2d}$ of dimension $2d-2$. For any affine interval exchange
transformation $T$, we denote by $\lambda(T)$ and $\rho(T)$ respectively

$\displaystyle\lambda(T)$
$\displaystyle:=(\lambda_{1},\dots,\lambda_{d})=(|I^{t}_{1}|,\dots,|I^{t}_{d}|),$
$\displaystyle\rho(T)$
$\displaystyle:=(\rho_{1},\dots,\rho_{d})=(DT_{1}(x_{1}),\dots,DT_{d}(x_{d})),\qquad
x_{j}\in I_{j}\,\quad\text{for\ all}\ 1\leq j\leq d,$

where $DT_{j}(x_{j}):=T_{j}^{\prime}(x)$ is the value of the derivative of the
branch $T_{j}$ of $T$ at any point $x_{j}$ in the interval $I_{j}$ and is
independent on the choice of $x_{j}$ since in an affine IET $T^{\prime}$ is
locally constant on $I_{j}$. We call $\lambda(T)$ the _length vector_ and
$\rho(T)$ the _slope vector_ of $T$.

####  2.2.3. Shape-profile coordinates for GIETs

We introduce now a set of coordinates on $\mathcal{X}_{\pi}^{r}$ which allow
us to endow $\mathcal{X}_{\pi}$ and consequently $\mathcal{X}^{r}$ with the
structure of a Banach manifold. These coordinates, that we will call _shape-
profile_ coordinates, where first introduced and used by the first author in
[Selim:loc] and will play a central role also in the present paper.

Let $T$ be a $\mathcal{C}^{r}$-GIET, with associated permutation $\pi$ and let
$(I_{i}^{t})_{1\leq i\leq d}$ and $(I_{i}^{b})_{1\leq i\leq d}$ be the _top_
and _bottom_ partitions of $[0,1]$ associated to it. We make the two following
observations.

  1. (1)

There is a unique affine interval exchange transformation $A_{T}$ mapping
$I_{i}^{t}$ to $I_{i}^{b}$.

  2. (2)

Furthermore, for all $1\leq i\leq d$, there is a unique element
$\varphi_{T}^{i}$ of $\mathrm{Diff}^{r}([0,1])$ such that the restriction of
$T$ to $I_{i}^{t}$ is equal to

$\varphi_{T}^{i}:=\mathcal{N}(T_{i}):=a_{i}\circ T_{i}\circ b_{i}$

where $b_{i}$ is the unique orientation preserving affine map mapping
$I_{i}^{t}$ onto $[0,1]$ and $a_{i}$ is the unique orientation preserving
affine map mapping $[0,1]$ onto $I_{\pi(i)}^{b}$.

Notice that, using these coordinates, if
$\rho=\rho(A_{T})=(\rho_{1},\dots,\rho_{d})$ is the slope vector of the shape
$A_{T}$, one has242424This follows from the explicit expression of the GIET in
terms of the profiles which is given by
$T(x)=u^{b}_{{\pi_{b}(i)-1}}+|I^{b}_{i}|\,\varphi^{i}_{T}(b_{i}(x)),\
\text{where}\
b_{i}(x)={(x-u^{t}_{\pi_{t}(i)-1})}/{|I^{t}_{i}|},\qquad\text{for\ all}\ x\in
I^{t}_{i}=(u^{t}_{\pi_{t}(i)-1},u^{t}_{\pi_{t}(i)}),\quad 1\leq i\leq d.$
noticing that $DA_{T}(x)=|I_{i}^{b}|/|I^{t}_{i}|$ for every $x\in I^{t}_{i}$
and therefore $\rho_{i}=|I_{i}^{b}|/|I^{t}_{i}|$.

(3) $DT(x)=DT_{i}(x)=\rho_{i}\,D\varphi^{i}_{T}(b_{i}(x)),\qquad\text{for\
all}\ x\in I^{t}_{i},\ 1\leq i\leq d,$

(where $b_{i}$ is as above the affine map which maps $I^{t}_{i}$ to $[0,1]$).

The operation of associated to a GIET a shape and a profile can be inverted
and therefore the map:

$T\longmapsto(A_{T},\varphi_{T}),\qquad\varphi_{T}:=(\varphi_{T}^{1},\cdots,\varphi_{T}^{d})$

gives an identification between $\mathcal{X}_{\pi}^{r}$ and
$\mathcal{A}_{\pi}\times(\mathrm{Diff}^{r}([0,1]))^{d}$ where
$\mathcal{A}_{\pi}$ the space of AIETs with permutation $\pi$. The AIET which
appears as projection onto the first coordinate, namely $A_{T}$, will be
called the _shape_ of $T$; the vector
$\varphi_{T}:=(\varphi_{T}^{1},\cdots,\varphi_{T}^{d})$ will be called
_profile_ of $T$.

We denote by $\mathcal{P}_{d}^{r}$ (or simply $\mathcal{P}_{d}$ or
$\mathcal{P}$ when the regularity $r$ or the number of intervals $d$ are not
relevant or clear from the context) the space
$\big{(}\mathrm{Diff}^{r}([0,1])\big{)}^{d}$ and so we have a canonical
identification

$\mathcal{X}_{\pi}^{r}=\mathcal{A}_{\pi}\times\mathcal{P}^{r}_{d},\qquad\mathcal{X}^{r}_{d}=\mathcal{A}_{d}\times\mathcal{P}^{r}_{d}=\bigcup_{\pi\in\mathfrak{S}^{0}}\mathcal{A}_{\pi}\times\mathcal{P}_{d}^{r}.$

Using this parametrisation, we can endow $\mathcal{X}_{d}^{r}$ with the
structure of a Banach manifold directly inherited from that of
$\mathrm{Diff}^{r}([0,1])$. Again, where there is no possible ambiguity, we
will drop the indexes $\pi$ and $r$ and simply write
$\mathcal{X}=\mathcal{A}\times\mathcal{P}$.

In the sequel we use the following notation for
$f:[0,1]\longrightarrow\mathbb{R}$ of class $\mathcal{C}^{r}$,

$||f||_{\mathcal{C}^{r}}=\max_{0\leq i\leq d}{||f^{(i)}||_{\infty}}$

where $f^{(i)}$ is the $i$-th derivative of $f$ and $||\cdot||_{\infty}$
denotes the sup norm. We extend this norm to
$(\mathcal{C}^{r}([0,1],\mathbb{R}))^{d}$ simply by taking the sum of the
norms on each coordinate.

###  2.3. Renormalization of GIETs

We introduce now the renormalization operator $\mathcal{V}$ on the space
$\mathcal{X}^{r}$ of GIET defined (on the subspace of GIETs with no
connections) by Rauzy-Veech induction. Rauzy-induction as a tool to study
standard IETs and their ergodic properties appears for the first time in the
seminal works by Rauzy [Ra:ech] and Veech [Ve:gau, Ve:inI] and has been a
standard tool in the theory since then (see e.g. [Zo:dev, AF:wea, Ul:mix,
Ul:abs, KKU, Ch:dis], $\dots$). Rauzy-Veech induction as a tool to study GIETs
and the notion of rotation number of a GIET appear e.g. in the works [MMY3,
MarmiYoccoz], see also [Yoc:Clay]. This section follows partly [MMY].

####  2.3.1. Elementary step of Rauzy-Veech induction

Let $T$ be a GIET on $d$ intervals (as in Definition 2.1.1). Consider the
partition endpoints $u^{t}_{j}$ and $u^{b}_{j}$, see § 2.1.4. Let
${\lambda}_{1}:=\max\\{u_{d-1}^{t},u_{d-1}^{b}\\}$. Thus $[0,\lambda_{1}]$ is
the interval $[0,1]\backslash I^{t}_{\pi_{t}(d)}$ if the last interval before
the exchange $I^{t}_{\pi_{t}(d)}$ is shorter than the last interval after the
exchange $I^{b}_{\pi^{t}(d)}$, while $[0,\lambda_{1}]=[0,1]\backslash
I^{b}_{\pi_{b}(d)}$ otherwise, i.e. if
$|I^{b}_{\pi_{b}(d)}|<|I^{t}_{\pi_{t}(d)}|$.

We define $T_{1}$ to be the first-return of $T$ on the interval
$[0,\lambda_{1}]$. One can verify that $T_{1}$ is well defined and is also a
GIET on $d$ intervals provided $u_{d-1}^{t}\neq u_{d-1}^{b}$. Define
$\mathcal{V}(T)$ to be $T_{1}$ normalised to be a map whose range is $[0,1]$:
formally $\mathcal{V}(T):=\mathcal{N}(T_{1})$ where $\mathcal{N}(T_{1})$ is
obtained conjugating $T_{1}$ by the unique affine map mapping $[0,l]$ to
$[0,1]$, namely

(4)
$\mathcal{V}(T):=\mathcal{N}(T_{1}),\qquad\text{where}\quad\mathcal{N}(T_{1})(x):=\frac{1}{\lambda_{1}}\,T_{1}\big{(}\lambda_{1}x\big{)},\qquad\textrm{for\
all}\ x\in[0,1].$

The operation consisting in passing from $T$ to $\mathcal{V}(T)$ is called the
elementary step of the Rauzy-Veech induction.

When one performs the elementary step of the induction, the associated
permutation $\pi$ changes. Note that the new permutation only depends on the
initial one and on whether $u_{d-1}^{t}>u_{d-1}^{b}$ or not. If
$u_{d-1}^{t}<u_{d-1}^{b}$ we say that the bottom interval wins and we say that
the top interval wins otherwise. Furthemore, we can record the label of the
interval which wins: if the interval which wins is $I^{t}_{j}$ (i.e. the top
interval wins and $I^{t}_{\pi_{t}(d)}=I^{t}_{j}$ we say that $j$ is the
_winner_). Similarly we say that $j$ is the _winner_ also if $I^{b}_{j}$ wins,
i.e. the bottom interval wins and $I^{b}_{\pi_{b}(d)}=I^{b}_{j}$.

One can show that if $T$ has no connections, also $T_{1}$ has no connections
and therefore it is possible to apply again an elementary step of the Rauzy-
Veech induction to $T_{1}$. Thus, if a GIET $T$ has no connections, Rauzy-
Veech induction can be iterated infinitely many times.

####  2.3.2. Paths on Rauzy diagrams and rotation numbers

We define now the notion of _rotation number_ associated to a GIET, see e.g.
[Yoc:CF, Yoc:Clay]. The rotation number will be an infinite path on a
combinatorial graph describing the moves of the renormalization algorithm (see
Defiintion 2.3.1).

We can form an oriented graph whose set of vertices is $\mathfrak{S}_{d}$ and
there is an oriented edge from one permutation $\pi_{1}$ to $\pi_{2}$ if and
only if there is a GIET with permutation $\pi_{1}$ whose image by the
elementary step of the Rauzy-Veech induction is a GIET with permutation
$\pi_{2}$. This oriented graph is called the _Rauzy diagram_. It has a certain
number of connected components (which were classified by Zorich in [Zo:Rau])
and are classically called _Rauzy classes_.

If a GIET has no connections, so then it is possible to iterate Rauzy-Veech
induction infinitely many times and get an infinite sequence
$T,\mathcal{V}(T),\mathcal{V}^{2}(T),\cdots,\mathcal{V}^{n}(T),\cdots$. For
every $n\in\mathbb{N}$ let $\pi_{n}$ be the combinatorial datum of
$\mathcal{V}^{n}(T)$ and let $\gamma_{n}$ be the arrow from $\pi_{n}$ to
$\pi_{n+1}$ which corresponds to the elementary step to pass from
$\mathcal{V}^{n}(T)$ to $\mathcal{V}^{n+1}(T)$. To the infinite orbit
$\\{\mathcal{V}^{2}(T),n\in\mathbb{N}\\}$ we can associate a path
$\gamma(T):=\gamma_{0}\gamma_{1}\cdots\gamma_{n}\cdots$ in the associated
Rauzy diagram passing through the vertices
$\pi_{0},\pi_{1},\cdots,\pi_{n},\cdots$ obtained concatenating the arrows
$\gamma_{n}$ describing the moves of the algorithm.

######  Definition 2.3.1 (Combinatorial rotation number).

Given a GIET n $T$ with no connections, its _combinatorial rotation number_
(or simply _rotation number_) is the datum of the Rauzy path
$\gamma(T)=\gamma_{1}\gamma_{2}\cdots\gamma_{n}\cdots$ associated to the orbit
$\\{\mathcal{V}^{n}(T),n\in\mathbb{N}\\}$.

The terminology rotation number252525We added the adjective _combinatorial_
since it gives a conjugacy invariant which describes the combinatorial
structure of orbits (as in other examples from one-dimensional dynamics, like
for example the _kneading sequences_ for unimodal maps) as well as to
distinguish it from other possible generalizations of the notion of rotation
number, such as _rotation vectors_ for higher dimensional tori or the _Katok
fundamental class_ which generalizes the _asymptotic cycle_ role also played
by the rotation number. (which was used in the works by Marmi-Moussa and
Yoccoz and advertised in the lecture notes by Yoccoz, see [Yoc:CF, Yoc:Clay])
has been chosen because for $d=2$, i. e. for GIETs with $d=2$ intervals, which
correspond to circle homeomorphisms, this piece of data is equivalent to the
datum of the usual rotation number. Furthermore, if $T_{1}$ and $T_{2}$ are
(semi)conjugate, then $\gamma(T_{1})=\gamma(T_{2})$, i.e. the rotation number
is an invariant of the (semi)conjugacy class. The converse (see Theorem 2.1
below) is also true for _irrational_ rotation numbers (to be defined below,
see Definition 2.3.2), a perhaps more important reason which further supports
that it is a good analogue of the classical rotation number.

####  2.3.3. Irrational combinatorial rotation numbers and semi-conjugacy with
a standard IET

If $T_{0}$ is a standard IET with no connection, its combinatorial rotation
number $\gamma(T_{0})$ has an additional property: all indexes $j$ in the
alphabet $\mathcal{A}=\\{1,\dots,d\\}$ of _labels_ of intervals are _winners_
infinitely many times. This property characterizes paths on the Rauzy diagram
which come from standard IETs (or GIETs which are semi-conjugated to standard
IETs, see above). In particular, if a path $\gamma$ on a Rauzy-diagram has
this property that every $j$ appears infinitely many times as a winner of an
arrow of $\gamma$ (a path on the Rauzy-diagram with this property is called
$\infty$-complete, see [MMY, Yoc:Clay, Yoc:cours]), then there exists262626The
IET is not necessarily unique, but any two standard IETs $T_{0}$ and $T_{1}$
with $\gamma(T_{0})=\gamma(T_{1})$ are topologically conjugated, see
[Yoc:cours]. a standard IET $T_{0}$ such that $\gamma(T_{0})=\gamma$.

Following [MMY, MMY3], we give the following definition. Let $T$ be a GIET
with no connection and $\gamma(T)$ its rotation number.

######  Definition 2.3.2 (irrational or ($\infty$-complete) rotation numbers).

The (combinatorial) rotation number $\gamma(T)$ is said to be _irrational_ (or
$\infty$-_complete_) iff every $j\in\mathcal{A}=\\{1,\dots,d\\}$ is the winner
of infinitely many arrows of $\gamma(T)$.

The following result, which is proved in the notes [Yoc:Clay] by Yoccoz,
extends Poincaré theorem for circle diffeomorphisms and further supports the
terminology ’rotation number’. We will refer to it as Poincaré theorem for
GIETs.

######  Theorem 2.1 (Poincaré theorem for GIETs, see [Yoc:CF, Yoc:Clay]).

Let $T$ be a GIET with $\infty$-complete rotation number and let $T_{0}$ be an
IET with same rotation number. Then $T$ is semi-conjugate to $T_{0}$.

In the rest of the paper, since we are interested in GIETs which are semi-
conjugated to a standard IET, we will always work GIETs with _irrational_
rotation number in the sense of Definition 2.3.2.

####  2.3.4. Periodic-type (or Fibonacci-type) combinatorics

We can now define also GIET of _periodic type_ (which are analogous to maps
with _Fibonacci type combinatorics_ in the one-dimensional dynamics
literature):

######  Definition 2.3.3 (Periodic type).

A GIET $T$ is called of _periodic type_ if it has no connections and its
combinatorial rotation number $\gamma(T)$ is _irrational_ and _periodic_ ,
i.e. there exists a $p>0$ such that $\gamma_{n+p}=\gamma_{n}$ for every
$n\in\mathbb{N}$. The minimal $p$ with such property will be called the
_period_ of $\gamma(T)$.

####  2.3.5. Definition of the renormalisation operator

The elementary step of the Rauzy-Veech induction can be used to define an
operator acting on an open subset of $\mathcal{X}^{r}$ defined the following
way. Set

$\mathcal{Y}^{r}=\\{T\in\mathcal{X}^{r}\ |\ u_{d-1}^{t}(T)=u_{d-1}^{b}(T)\\}.$

Note that $\mathcal{Y}^{r}$ is a codimension $1$ smooth submanifold of
$\mathcal{X}^{r}$. In other words, $\mathcal{Y}^{r}$ is the subset of those
GIETs for which the rightmost top and bottom intervals have same length. It is
exactly the set for which the elementary step of the Rauzy-Veech induction is
not defined. Thus the elementary step of the Rauzy-Veech induction defines an
operator
$\mathcal{V}:\mathcal{X}^{r}\setminus\mathcal{Y}^{r}\mapsto\mathcal{X}^{r}$
given by $T\longmapsto\mathcal{V}(T)$.

A GIET $T$ is said to be infinitely renormalizable iff $\mathcal{V}^{n}(T)$ if
well-defined for all $n\in\mathbb{N}$. Note that in particular a GIET with
irrational (i.e. $\infty$-complete, see Definition 2.3.2) rotation number is
infinitely renormalizable. However, not all infinitely renormalizable GIETs
have irrational ($\infty$-complete) rotation number (actually in parameter
space, the "generic case" is expected not to have irrational combinatorial
rotation number).

####  2.3.6. Accelerations

We will consider as renormalization operators $\mathcal{R}$ operators that are
obtained _accelerating_ $\mathcal{V}$, i.e. such that
$\mathcal{R}^{k}(T):=\mathcal{V}^{n_{k}}(T)$ where is a suitably chosen
sequence of iterates of $\mathcal{V}$ (which depends on $T$). For example,
when $T$ is of _periodic type_ with period $p$ (see Definition 2.3.3), the
natural renormalization operator to use is simply
$\mathcal{R}:=\mathcal{V}^{p}$, so that
$\mathcal{R}^{k}(T):=\mathcal{V}^{kp}(T)$ for every $k\in\mathbb{N}$. A
classical acceleration of $\mathcal{V}$ is the _Zorich acceleration_ , which
we will denote by $\mathcal{Z}$ and corresponds to grouping together all
successive elementary steps of Rauzy-Veech induction which are equal of type
top, or bottom, respectively: given $T$ with irrational $\gamma(T)$, one can
show that top and bottom both win infinitely often; therefore, one can define
the sequence $(n_{k})_{k\in\mathbb{N}}$ such that
$\mathcal{Z}^{k}(T):=\mathcal{V}^{n_{k}}(T)$ by setting $n_{0}:=0$ and,
recursively, if $n_{k}$ is such the top (resp. bottom) interval of
$\mathcal{V}^{n_{k}}(T)$ wins, setting $n_{k+1}$ to be the first $n>n_{k}$
such that bottom (resp. top) interval of $\mathcal{V}^{n}(T)$ wins.

Accelerations can also be obtained considering _inducing_ (ie. first return
maps): if $E\subset\mathcal{X}^{r}$ is a subset, we can obtain an acceleration
of $\mathcal{V}$, denoted by $\mathcal{V}_{E}$, defined on the set of
$T\in\mathcal{X}^{r}$ which visit $E$ infinitely often: if
$(n_{k})_{k\in\mathbb{N}}$ is the sequence of successive visits of the orbit
$(\mathcal{V}^{n}(T))_{n\in\mathbb{N}}$ to $T$ (i.e. we set $n_{1}$ to be the
first $n\geq 0$ such that $\mathcal{V}^{n}(T)\in E$ and, given $n_{k}$, we set
$n_{k+1}$ to be the smallest $n>n_{k}$ such that $\mathcal{V}^{n}(T)\in E$),
we can defined $\mathcal{V}_{E}^{k}(T):=\mathcal{V}^{n_{k}}(T)$.

####  2.3.7. Dynamical partitions

We introduce the notion of dynamical partitions. Let $T$ be an infinitely
renormalizable $T$ and let $\mathcal{R}$ be a renormalization operator
obtained by accelerating Rauzy-Veech induction, as described above. Then the
orbit $(\mathcal{R}^{n}(T))_{n\in\mathbb{N}}$ is well defined and, by
definition, for every $n\in\mathbb{N}$ the GIET $\mathcal{R}^{n}(T)$ is
obtained by rescaling the first return map of $T$ on an interval of the form
$[0,\lambda_{n}]$. We will denote by $I^{(n)}:=[0,\lambda_{n}]$ and by $T_{n}$
the Poincaré map of $T$ to $I^{(n)}$, so that
$\mathcal{R}^{n}(T)(x)=T_{n}(\lambda_{n}x)/\lambda_{n}$ or, explicitely, if
$I^{t}_{j}(n)$ denote the continuity intervals for $\mathcal{R}^{n}(T)$,

(5)
$\mathcal{R}^{n}(T)(x)=\frac{T_{n}(\lambda_{n}x)}{\lambda_{n}}=\frac{T^{q^{(n)}_{j}}(\lambda_{n}x)}{\lambda_{n}},\qquad\mathrm{for\
all}\ x\in I^{t}_{j}(n)=\frac{1}{\lambda_{n}}I^{(n)}_{j}.$

Notice that $\\{I^{(n)},\ n\in\mathbb{N}\\}$ are nested intervals with $0$ as
a common left endpoint. By construction $T_{n}$ is a $d-$GIET. We denote by
$I^{(n)}_{j}$, for $j=1,\dots,d$ its continuity intervals, so that the
interval $I^{(n)}=[0,\lambda_{n}]$ is partitioned into
$I^{(n)}=\cup_{j=1}^{d}{I^{(n)}_{j}}$ and for each $1\leq j\leq d$, $T_{n}$
restricted to $I^{(n)}_{j}$ is equal to $T^{q^{(n)}_{j}}$ where $q_{j}^{(n)}$
is the first return time of $I^{(n)}_{j}$ to $I^{(n)}$ under $T$, i.e. the
minimum $q\geq 1$ such that $T^{q}(x)\in I^{(n)}$ for some (hence all) $x\in
I^{(n)}_{j}$.

Let us define

$\mathcal{P}_{n}:=\bigcup_{j=1}^{d}{\mathcal{P}^{j}_{n}},\qquad\text{where}\quad\mathcal{P}^{j}_{n}:=\\{I^{(n)}_{j},T(I^{(n)}_{j}),T^{2}(I^{(n)}_{j}),\cdots,T^{q^{(n-1)}_{j}}(I^{(n)}_{j})\\}.$

One can verify that $\mathcal{P}_{n}$ is a partition of $[0,1]$ into
subintervals and that, for each $1\leq j\leq d$, the collection
$\mathcal{P}^{j}_{n}$ is a _Rohlin tower_ by intervals, i.e. a collection of
disjoint intervals which are mapped one into the next by the action of $T$.

We will say that $\\{\mathcal{P}_{n},\ n\in\mathbb{N}\\}$ is the sequence of
_dynamical partitions_ _associated to_ the orbit
$(\mathcal{R}^{n})_{n\in\mathbb{N}}$ of $T$ under the renormalization operator
$\mathcal{R}$ and, when the renormalization operator is clear, that
$\mathcal{P}_{n}$ is the dynamical partition _of level_ $n$. We say that the
number $q_{j}^{(n)}$ of intervals in a tower is the _height_ of the (Rohlin)
tower $\mathcal{P}^{j}_{n}$. Thus, $\mathcal{P}_{n}$ also gives a
representation of $[0,1]$ as a _skyscraper_ , i.e. a collection of Rohlin
towers, for $T$. Notice that if $n>m$, then the partition $\mathcal{P}_{n}$ is
a refinement of $\mathcal{P}_{m}$.

###  2.4. One dimensional dynamics toolkit

We recall here classical and crucial tools in the theory of circle
diffeomorphisms and more in general in one-dimensional dynamics, such as non-
linearity and Schwarzian derivative.

####  2.4.1. Non-linearity

For any $\mathcal{C}^{2}$ map $f:I\longrightarrow J$ where $I$ and $J$ are
open intervals such that $\mathrm{D}f$ does not vanish, one can define the
_non-linearity_ function $\eta_{f}$ to be the function
$\eta_{f}:I\to\mathbb{R}$ given by

$\eta_{f}(x):=(\mathrm{D}\log\mathrm{D}f)(x)=\frac{\mathrm{D}^{2}f(x)}{\mathrm{D}f(x)}.$

The function $\eta_{f}$ is called non-linearity since it measures how far $f$
is from being affine and has the property that $\eta_{f}\equiv 0$ if and only
if $f$ is an affine map. Some easy but important consequence for the non-
linearity are the following.

######  Lemma 2.4.1 (properties of non-linearity).

Let $f:I\longrightarrow J$, $g:J\longrightarrow K$ be diffeomorphisms of class
$\mathcal{C}^{2}$. Then the following properties hold:

  1. (i)

_Chain rule for non-linearity_ : $\,\eta_{g\circ
f}=\eta_{f}+\mathrm{D}g\,(\eta_{g}\circ f)$;

  2. (ii)

_Distribution property_ : $\,\int_{I}{\eta_{g\circ
f}}=\int_{I}{\eta_{f}}+\int_{J}{\eta_{g}}$;

  3. (iii)

_Triangular inequality_ : $\,\int_{I}{|\eta_{g\circ
f}|}\leq\int_{I}{|\eta_{f}|}+\int_{J}{|\eta_{g}|}$.

The second and third points are consequence of the first, which itself is an
application of the chain-rule for differentiable functions. We refer the
reader for example to the Appendix of [Martens] for more details.

######  Definition 2.4.1 (mean and total non-linearity of a GIET).

Given a $\mathcal{C}^{2}$ interval exchange map $T:[0,1]\to[0,1]$, defined on
the continuity intervals $I_{j}\subset[0,1]$ we define the non-linearity
$\eta_{T}$ to be the (bounded) piecewise continuous map from $[0,1]$ to
$\mathbb{R}$ given by

$\eta_{T}(x):=\eta_{T_{j}}(x),\qquad\mathrm{if}\ x\in I_{j},\quad 1\leq j\leq
d,$

where $T_{j}:I_{j}\to[0,1]$ are the branches of $T$ obtained restricting $T$
to its continuity intervals. We subsequently define

$\overline{N}(T):=\int_{0}^{1}{\eta_{T}(x)dx},\qquad|N|(T):=\int_{0}^{1}{|\eta_{T}(x)|dx}.$

We call $\overline{N}(T)$ the _mean non-linearity_ of $T$ and $|N|(T)$ the
_total non-linearity_ of $T$.

Mean non-linearity and total non-linearity play an important role in the
theory of renormalization, in particular since, seen as functions
$\overline{N}(\cdot)$ and $|N|(\cdot)$ on the space $\chi^{r}$ of GIET (with
$r\geq 2$) they satisfies the properties listed in the following proposition.

######  Proposition 2.4.1 (properties of mean and total non-linearity).

For any $r\geq 2$, $\pi\in\mathfrak{S}_{r}$, the mean non-linearity
$\overline{N}(\cdot)$ and the total non-linearity $|N|(\cdot)$ have the
following properties:

  * (i)

$|\overline{N}(T)|\leq|N|(T)$ for every $T\in\mathcal{X}^{r}$;

  * (ii)

$\overline{N}(\cdot)$ is invariant under renormalization, i.e.
$\overline{N}(\mathcal{V}(T))=\overline{N}(T)$ for every
$T\in\mathcal{X}^{r}$;

  * (iii)

$|N|(\cdot)$ is decreasing under renormalization, i.e.
$|N|(\mathcal{V}(T))\leq|N|(T)$ for every $T\in\mathcal{X}^{r}$;

  * (iv)

$\overline{N}(T)$ is invariant under rescaling by (restrictions) of affine
maps, so that in particular if $a,b$ are (restrictions of) linear maps,
$\overline{N}(a\circ T\circ b)=\overline{N}(T)$.

###### Proof.

Note that the branches of $\mathcal{V}(T)$ are compositions of restrictions of
branches of $T$. These properties are thus easy consequences of Properties
$(ii)$ and $(iii)$ of Lemma 2.4.1 applied to branches of $\mathcal{V}(T)$. ∎

In light of $(iii)$, the total non-linearity $|N|(T)$ captures how close $T$
is to the set of affine interval maps (see in particular Remark 4.2.1). This
will play a central role in proofs of convergence of renormalization.

####  2.4.2. Distortion bounds

We now recall standard distortion bounds in one-dimensional dynamics and
derive from it an important consequence for the renormalization operator
$\mathcal{R}$.

######  Lemma 2.4.2 (Distorsion bound).

Let $T$ be a GIET of class $\mathcal{C}^{2}$. Let $J\subset[0,1]$ be an
interval such that $J,T(J),T^{2}(J),\cdots,T^{n}(J)$ are pairwise disjoint and
do not contain any singularities of $T$. Then we have

$\left|\frac{\mathrm{D}(T^{n})(x)}{\mathrm{D}(T^{n})(y)}\right|\leq\exp{|N|(T)}:=\exp\left(\int_{0}^{1}{|\eta_{T}(x)|dx}\right),\qquad\text{for\
all}\ x,y\in J.$

The proof is an adaptation to GIETs of the classical proof. We included it in
Appendix A.2 for complenetess.

####  2.4.3. The Schwarzian derivative

We conclude this Section by introducing the Schwarzian derivative which is a
most classical tool in one-dimensional dynamics. If $f:I\longrightarrow J$ is
a $\mathcal{C}^{3}$ diffemorphism between two connected intervals $I$ and $J$,
define its Schwarzian derivative to be

$\mathrm{S}(f):=\frac{\mathrm{D}^{3}f}{\mathrm{D}f}-\frac{3}{2}\left(\frac{\mathrm{D}^{2}f}{\mathrm{D}f}\right)^{2}.$

Non-linearity and Schwarzian derivative are related by the following
equivalent expression for $\mathrm{S}(f)$:

(6) $\mathrm{S}(f)=\mathrm{D}\eta_{f}-\frac{1}{2}\eta_{f}^{2}.$

The Schwarzian derivative enjoys the two following important properties :

  * (S1)

$\mathrm{S}\left(f\right)$ identically vanishes if and only if $f$ is the
restriction of a Moebius map to its domain.

  * (S2)

if $f:I_{1}\to I_{2}$ and $g:I_{2}\to I_{3}$, the composition $g\circ
f:I_{1}\to I_{3}$ satisfies the following _chain rule for the Schwarzian
derivative_ :

$\mathrm{S}\left(g\circ f\right)=\mathrm{S}(g)\circ
f\,(\mathrm{D}f)^{2}+\mathrm{S}(f).$

For $f:I\rightarrow J$, where $I$ and $J$ are intervals, we denote by
$\mathcal{N}(f)$ the normalisation of $f$ given by $\mathcal{N}(f):=b\circ
f\circ a$, where $a$ and $b$ are respectively the only orientation-preserving
affine map mapping $[0,1]$ onto $I$ and $J$ onto $[0,1]$. Then, from $(S1)$
and $(S2)$ we can deduce272727Property $(S3)$ follows since by $(S1)$ we have
that $\mathrm{S}(a)=\mathrm{S}(b)=0$ and thus, by $(S2)$,
$\mathrm{S}\left(b\circ f\circ a\right)=\mathrm{S}\left(f\circ
a\right)=\left(\mathrm{S}(f)\circ a\right)(a^{\prime})^{2}$, which gives the
desired conclusion since $a^{\prime}(x)=|I|$ for all $x\in[0,1]$. that

  * (S3)

$\mathrm{S}\left(\mathcal{N}(f)\right)=\mathrm{S}\left(b\circ f\circ
a\right)=|I|^{2}\,\mathrm{S}(f)\circ a$.

###  2.5. The Zorich cocycle

In this section we restrict the Rauzy-Veech renormalization $\mathcal{V}$
(which was defined in § 2.3 for GIETs in
$\mathcal{X}^{r}\backslash\mathcal{Y}^{r}$) to the subset
$\mathcal{I}_{d}\subset\mathcal{X}^{r}$ of _standard_ IETs. This is the
classical setup in which the induction was introduced by Rauzy and Veech
[Ra:ech, Ve:gau] and studied from the ergodic theory point of view. We
introduce the Rauzy-Veech and Zorich cocycle and recall the integrability
property and Oseledets theorem for the latter.

####  2.5.1. Invariant measures

Let us recall that $\mathcal{I}_{d}$ is isomorphic to
$\Delta_{d-1}\times\mathfrak{S}^{0}_{d}$ (refer to § 2.2.1). A natural measure
on $\mathcal{I}_{d}$, which we will call _Lebesgue measure_ , is the product
measure obtained taking the product of the Lebesgue measure on
$\mathbb{R}^{d}$ restricted to the simplex $\Delta_{d-1}$ and the counting
measure 282828The counting measure $\delta$ on $\mathfrak{S}^{0}_{d}$ is
simply the measure defined by setting $\delta(S)$ to be the cardinality of $S$
for any subset $S\subset\mathfrak{S}^{0}_{d}$. It is here simply used to put a
copy of Lebesgue measure on each _copy_ of the simplex
$\Delta_{d}\times\\{\pi\\}$ indexed by $\pi\in\mathfrak{S}^{0}_{d}$. the
measure defined, for any $\pi_{1},\pi_{2}\in\mathfrak{S}^{0}_{d}$, by
$\delta(\pi_{1},\pi_{2})=1$ iff $\pi_{1}=\pi_{2}$ and $0$ otherwise. Thus, for
any $0<\epsilon<1$, asking that $d_{\mathcal{C}^{1}}(T_{1},T_{2})<\epsilon$,
where, for $i=1,2$, $T_{i}$ is a GIET with combinatorial datum $\pi_{i}$ and
shape-profile coordinates $(A_{T_{i}},\varphi_{T_{i}})$, is equivalent to
asking that $\pi_{1}=\pi_{2}$, $d_{\mathcal{A}}(T_{1},T_{2})<\epsilon$ and
$d_{\mathcal{C}^{1}}^{\mathcal{P}}(\varphi_{T_{1}},\varphi_{T_{2}})<\epsilon$.
on combinatorial data $\mathfrak{S}^{0}_{d}$. We refer to its measure
class292929Recall that a _measure class_ is an equivalence class of measures
which have the same sets of measure zero. as _Lebesgue measure class_.

The domain of definition of an elementary step of Rauzy-Veech induction
$\mathcal{V}$ (acting on standard IETs) is hence

$\mathcal{I}_{d}:=\mathcal{I}_{d}\backslash\mathcal{Y}^{r}=\\{T=(\pi,\lambda)\in\mathcal{I}_{d},\,\,\text{such\
that\ }u^{t}_{d-1}\neq
u^{b}_{d-1}\\}=\\{T=(\pi,\lambda)\in\mathcal{I}_{d},\,\,\text{such\ that\
}\lambda_{\pi_{t}(d)}\neq\lambda_{\pi_{b}(d)}\\}$

and therefore it is a full measure subset of $\mathcal{I}_{d}$ with respect to
the Lebesgue measure class (defined above).

Le us fix an irreducible $\pi\in\mathfrak{S}_{d}^{0}$ (see § 2.1.3) and
consider the action of $\mathcal{V}$ restricted to the space
$\mathcal{I}_{\pi}:=\Delta_{d-1}\times\mathcal{R}(\pi)$, where
$\mathcal{R}(\pi)$ is the Rauzy class303030Let us recall that the Rauzy class
of $\pi$ is the subset of all permutations $\pi^{\prime}$ of $d$ symbols which
appear as permutations of an IET
$T^{\prime}=(\underline{\lambda}^{\prime},\pi^{\prime})$ in the orbit under
$\mathcal{R}$ of some IET $(\underline{\lambda}^{\prime},\pi)$ with initial
permutation $\pi$. of $\pi$ (see § 2.3.1). Veech proved in [Ve:gau] is that
the restriction of $\mathcal{V}:\mathcal{I}_{\pi}\to\mathcal{I}_{\pi}$ admits
an invariant measure which is absolutely continuous with respect to the
Lebesgue measure on $\mathcal{I}_{d}$ (see above), but which is infinite.
Dropping the dependence on $\pi$ (or more precisely on the Rauzy class
$\mathcal{R}(\pi)$) we will denote this natural measure by $\mu_{\mathcal{V}}$
when $\pi$ is fixed. This seminal result started the study of $\mathcal{V}$
from the ergodic theoretical point of view. Veech also showed already in
[Ve:gau] that $\mathcal{V}:\mathcal{I}_{\pi}\to\mathcal{I}_{\pi}$ is
conservative and ergodic with respect to $\mu_{\mathcal{V}}$.

The acceleration $\mathcal{Z}$ defined by Zorich was introduced to have a
_finite_ invariant measure: in [Zo:gau] Zorich showed indeed that
$\mathcal{Z}$ is defined on a full measure set of $\mathcal{I}_{\pi}$ and
admits a _finite_ invariant measure that we will denote $\mu_{\mathcal{Z}}$.
It follows from the definition and [Ve:gau] that also $\mathcal{Z}$ is ergodic
with respect to $\mu_{\mathcal{Z}}$.

######  Remark 2.5.1.

Since both $\mu_{\mathcal{Z}}$ and $\mu_{\mathcal{V}}$ are _absolutely
continuous_ with respect to the Lebesgue measure and
$\mathcal{I}_{d}=\cup_{\pi\in\mathfrak{S}_{d}^{0}}\mathcal{I}_{\pi}$, to show
that a property holds for a full measure set of IETs for the Lebesgue measure
on $\mathcal{I}_{d}$ it is sufficient to prove that, for any fixed irreducible
combinatorial datum $\pi\in\mathfrak{S}_{d}^{0}$, the it holds for
$\mu_{\mathcal{Z}}$ (or $\mu_{\mathcal{Z}}$) almost every $T$ in
$\mathcal{I}_{\pi}$.

####  2.5.2. Natural extension.

Notice that Rauzy-Veech induction $\mathcal{V}$ is not injective
($\mathcal{V}$ is actually two-to-one) and therefore neither $\mathcal{V}$ nor
its Zorich acceleration $\mathcal{Z}$ are _invertible_ on the space
$\mathcal{I}_{d}$. One can consider, though, its _natural extension_
$\hat{\mathcal{Z}}$ defined on the measure space
$(\hat{\mathcal{I}}_{\pi},\mu_{\hat{\mathcal{Z}}})$: this is a map such that
$\hat{\mathcal{Z}}$ is defined and _invertible_ on a full measure subset of
$\hat{\mathcal{I}}_{\pi}$ with respect to the measure
$\mu_{\hat{\mathcal{Z}}}$, and is an _extension_ of $\mathcal{Z}$ in the sense
of ergodic theory, i.e. there exists a projection
${p}:\hat{\mathcal{I}}_{\pi}\to{\mathcal{I}}_{\pi}$ such that
$\hat{p}\hat{\mathcal{Z}}=\mathcal{Z}\circ{p}$ (i.e. $p$ intertwines the
dynamics of $\mathcal{Z}$ and $\hat{\mathcal{Z}}$) and the measure
${m}_{\hat{\mathcal{Z}}}$ is the pull-back of ${m}_{\hat{\mathcal{Z}}}$ via
$p$, i.e. for every measurable set $E\subset\mathcal{I}_{\pi}$,
${m}_{\hat{\mathcal{Z}}}(p^{-1}(E))={m}_{{\mathcal{Z}}}(E)$. Notice that the
invariant measure $m_{\hat{\mathcal{Z}}}$ preserved by the map
$\hat{\mathcal{Z}}$ is also finite.

One can describe explicitely a geometric realization of these natural
extensions and the space $\hat{\mathcal{I}}_{\pi}$ can be identified with the
the space of _zippered rectangles_ introduced by Veech in [Ve:gau] (consisting
of triples $\hat{T}=(\pi,\lambda,\tau)$ where $T=(\pi,\lambda)$ is a standard
IET and $\tau$ is a _suspension datum_ which contains the information required
to define a translation surface which has $T$ as Poincaré map, as in § 2.1.6).
We will not make explicit use of this interpretation, so we will simply denote
by $\hat{T}$ a point of $\hat{\mathcal{I}_{d}}$ such that $p(\hat{T})=T$ (here
if $\hat{T}=(\pi,\lambda,\tau)$, $p(\hat{T})=T$ is the IET $T=(\pi,\lambda)$
obtained forgetting the suspension datum $\tau$).

####  2.5.3. Basics on cocycles.

We recall now basic definitions concerning cocycles and their accelerations.
We refer the reader for example to [ViaLP] for a comprehensive introduction to
cocycles and Lyapunov exponents. Let $(X,\mu,F)$ be a discrete dynamical
system, where $(X,\mu)$ is a probability space and $F$ is a $\mu$-measure
preserving map on $X$. A a measurable map $A:X\rightarrow SL(d,\mathbb{C})$
($d\times d$ invertible matrices) determines a cocycle $A$ on $(X,\mu,F)$. If
we denote by $A_{n}(x)=A(F^{n}x)$ and by $A_{F}^{n}(x)=A_{n-1}(x)\cdots
A_{1}(x)A_{0}(x)$, the following _cocycle identity_

(7) $A_{F}^{m+n}(x)=A_{F}^{m}(F^{n}x)A_{F}^{n}(x)$

holds for all $m,n\in\mathbb{N}$ and for all $x\in X$. If $F$ is invertible,
let us set $A_{-n}(x)=A(F^{-n}x)$. The map $A^{-1}(x)=A(x)^{-1}$ gives a
cocycle over $F^{-1}$ which we call _inverse cocycle_. Let us set
$A_{-n}(x)=A(F^{-n}x)$. for $n<0$ we can set
$A^{(-n)}(x)=A^{-1}(F^{-1}x)\cdots A^{-1}(F^{-n}x)$, so that (7) holds for all
$n,m\in\mathbb{Z}$. Remark that $A^{(-n)}(x)=(A^{(n)}(T^{-n}x))^{-1}$. The
_inverse transpose cocycle_ $(A^{-1})^{T}$ is defined by
$(A^{-1})^{T}(x)=(A^{-1})^{T}(x)$ where $M^{T}$ denotes the transpose of $M$.

####  2.5.4. Induced cocycles and accelerations.

If $Y\subset X$ is a measurable subset, the induced map (or first return map,
or Poincaré map) of $F$ on $Y$, which is defined $\mu$-almost everywhere by
Poincaré recurrence, is the map given by $F^{r_{Y}(y)}(y)$ where
$r_{Y}(y):=\min\\{r\,|\,\,F^{r}y\in Y\\}$. The _induced cocycle_ $A_{Y}$ on
$Y$ is a cocycle over $(Y,\mu_{Y},F_{Y})$ where $F_{Y}$ is the induced map of
$F$ on $Y$ and $\mu_{Y}=\mu/\mu(Y)$ and $A_{Y}(y)$ is defined for all $y\in Y$
which return to $Y$ and is given by

$A_{Y}(y)=A(F^{r_{Y}(y)-1}y)\cdots A\left(Fy\right)A\left(y\right),$

where $r_{Y}(y)$ is again the first return time.

The induced cocycle is an _acceleration_ of the original cocycle, i.e. if
$\\{n_{k}\\}_{k\in\mathbb{N}}$ is the infinite sequence of return times of
some $y\in Y$ to $Y$ (i.e. $T^{n}y\in Y$ iff $n=n_{k}$ for some
$k\in\mathbb{N}$ and $n_{k+1}>n_{k}$) then

(8) $(A_{Y})_{k}(y)=A_{n_{k+1}-1}(y)\cdots A_{n_{k}+1}(y)A_{n_{k}}(y).$

We say that $x\in X$ is _recurrent_ to $Y$ under $T$ if there exists an
infinite increasing sequence $\\{n_{k}\\}_{k\in\mathbb{N}}$ such that
$T^{n_{k}}x\in Y$. Let us extend the definition of the induced cocycle $A_{Y}$
to all $x\in X$ recurrent to $Y$. If the sequence
$\\{n_{k}\\}_{k\in\mathbb{N}}$ is increasing and contains all
$n\in\mathbb{N}^{+}$ such that $T^{n}x\in Y$, let us say that $x$ _recurs to_
$Y$ _along_ $\\{n_{k}\\}_{k\in\mathbb{N}}$. In this case, let us set

$A_{Y}(x):=A(y)A^{n_{0}}_{F}(x),\quad\mathrm{where}\,\,y:=F^{n_{0}}x\in
Y;\qquad(A_{Y})_{n}(x):=(A_{Y})_{n}(y),\qquad\mathrm{for}\,\,n\in\mathbb{N}^{+}.$

If $F$ is ergodic, $\mu$-a.e. $x\in X$ is recurrent to $Y$ and hence $A_{Y}$
is defined on a full measure set of $X$.

####  2.5.5. Integrability

Here and in the rest of the paper, we will use the norm
$\|A\|=\sum_{ij}|A_{ij}|$ on matrices (more generally), the same results on
cocycles hold for any norm on $SL(d,\mathbb{Z})$). Remark that with this
choice $\|A\|=\|A^{T}\|$.

A cocycle over $(X,F,\mu)$ is called _integrable_ if
$\int_{X}\ln\|A(x)\|\mathrm{d}\mu(x)<\infty$. Integrability is the assumption
which allows to apply _Oseledets Theorem_ , also known as _multiplicative
ergodic theorem_ (the conclusion of Oseledets theorem in the setting of the
Zorich cocycle is recalled in § 2.5.10; for a more general reference, see e.g.
Section 4 in [ViaLP]). If $A$ is an integrable cocycle over $(X,F,\mu)$
assuming values in $SL(d,\mathbb{Z})$, then one can show that the dual cocycle
$(A^{-1})^{T}$ and, if $F$ is invertible, the inverse cocycle $A^{-1}$ over
$(X,F^{-1},\mu)$ are integrable. Furthermore, any induced cocycle $A_{Y}$ of
$A$ on a measurable subset $Y\subset X$ is integrable (see for example
[ViaLP], § 4.4.1).

####  2.5.6. The Zorich cocycle.

Consider the Zorich map $\mathcal{Z}$ on $\mathcal{I}_{d}$. Given
$T=(\lambda,\pi)\in\mathcal{I}_{d}$ with no connections and irreducible $\pi$,
denote by $\\{I^{(k)},\,{k\in\mathbb{N}}\\}$ the sequence of inducing
intervals corresponding to Zorich acceleration $\mathcal{Z}$ of the Rauzy-
Veech algorithm $\mathcal{V}$. Write
$\mathcal{Z}^{k}(T):=(\pi^{(k)},\lambda^{(k)})$ and let $T_{k}$ be the (non
normalised) induced IET, given by first returns of $T$ to $I^{(k)}$, so that

$\mathcal{Z}^{k}(T)=(\pi^{(k)},\underline{\lambda}^{(k)}/|I^{(k)}|),\quad\text{where}\
|I^{(k)}|=\sum_{j=1}^{d}\lambda^{(k)}_{j}=|\lambda^{(k)}|.$

Recall from § 2.3.7 that $T$ can be represented as skyscraper over $I^{(k)}$
and let

$q^{(k)}=q^{(k)}(T):=\left(q_{1}^{(k)},\dots,q_{d}^{(k)}\right)^{T},$

be the column vector whose entries $q^{(k)}_{j}$ are the heights of the Rohlin
towers, the vector of _heights_ or, equivalently, the vector of first return
times, since $q^{(k)}_{j}$ is also the first return time of $I^{(k)}_{j}$ to
$I$.

For each $T=T^{(0)}$ for which
$\mathcal{Z}(T)=(\pi^{(1)},\underline{\lambda}^{(1)}/|I^{(1)}|)$ is defined,
let us associate to $T$ the matrix $Z=Z(T)$ in $SL(d,\mathbb{Z})$ such that
$q^{(1)}=Z\,q^{(0)}$. The map $Z$: $X\rightarrow SL(d,\mathbb{Z})$ is a
cocycle over $(X,\mu_{\mathcal{Z}},\mathcal{Z})$, which we call the _Zorich
cocycle_ 313131Notice that there is also another cocycle, also sometimes
called Zorich cocycle, which transforms _lengths_ and is actually the
transpose inverse of the cocycle here defined. (also sometimes called
_Kontsevich-Zorich_ cocycle). Explicitely, if $T$ has combinatorial datum
$\pi=(\pi_{t},\pi_{b})$ and lengths vector $\lambda$, the cocycle $Z=Z(T)$ is
given by

$Z=Z(T)=\begin{cases}I_{d}+E_{\pi_{b}(d)\pi_{t}(d)}&\qquad\text{if}\,\lambda_{\pi_{t}(d)}>\lambda_{\pi_{b}(d)}\quad\text{(ie.\leavevmode\nobreak\
top\ is\ winner)},\\\
I_{d}+E_{\pi_{t}(d)\pi_{b}(d)}&\qquad\text{if}\,\lambda_{\pi_{t}(d)}<\lambda_{\pi_{b}(d)}\quad\text{(i.e.\leavevmode\nobreak\
bottom\ is\ winner)},\end{cases}$

where $I_{d}$ denotes the $d\times d$ identity matrix, while $E_{ij}$ denotes
the matrix which has all entries equal to zero, but the entry $ij$ which is
equal to $1$.

Zorich proved in [Zo:gau] that $Z$ is _integrable_. Defining:

$Z_{n}=Z_{n}(T):=Z(\mathcal{Z}^{n}(T)),\qquad Z^{(n)}:=Z_{n-1}\cdot\dots\cdot
Z_{1}Z_{0}$

and iterating the above relation, we then get the following matrix product
relation:

(9) $\underline{q}^{(n)}=Z^{(n)}q^{(0)},\qquad\mathrm{where}\quad
q^{(0)}(T):=\left(1,\dots,1\right)^{T},$

which gives the heights (i.e. return times) of the representation of $T$ as a
skyscraper over the firt return map $T_{n}$ to $I^{(n)}$. For more general
products with $m<n$ we use the notation

(10) $Q{(m,n)}\doteqdot Z_{n-1}Z_{n-2}\,\dots\,Z_{m+1}Z_{m}.$

The following _cocycle relation_ then holds for any triple of integers
$n,m,p$:

(11) $Q(n,p)=Q(m,p)\,Q(n,m),\qquad\text{for\ all}\ n<m<p.$

Notice that by choice of the norm $|q|=\sum_{j}|q_{j}|$ on vectors and
$\|A\|=\sum_{i,j}|A_{ij}|$ on matrices and since return times are positive
numbers,

(12)
$\max_{j}q^{(n)}_{j}\leq|q^{(n)}|\leq\|Q{(m,n)}\|\,\|q^{(m)}\|,\qquad\text{for\
any}\ m<n.$

####  2.5.7. Dynamical interpretation of the entries.

The _entries_ of the Zorich cocycle matrices have the following crucial
dynamical interpretation: if $T^{(n)}$ is the sequence of IETs obtained
inducing on the sequence $I^{(n)},n\in\mathbb{N}$ of intervals given by the
Zorich acceleration (so that $\mathcal{Z}^{n}(T)$ is obtained normalising
$T^{(n)}$ to an interval exchange acting on the unit interval), then the entry
$(Z_{n})_{ij}$ of the $n^{th}$ Zorich matrix $Z_{n}$ gives the number of
visits of the orbit of $x\in I^{(n+1)}_{j}$ under $T^{(n)}$ to $I^{(n)}_{i}$
up to its first return to $I^{(n+1)}$.

Correspondingly, the Zorich cocycle has also an interpretation in terms of
_incidence matrices_ of Rohlin towers (defined in § 2.3.7). The Rohlin towers
at step $n+1$ can be obtained by a _cutting and stacking_ 323232We do not give
here a precise definition of _cutting and stacking_ , which is a standard
construction in the study of ergodic theory and in particular of _rank one_
and, more in general, _finite rank_ dynamical systems., which construction
from the Rohlin towers at step $n$: more precisely, for any $n\in\mathbb{N}$
and $1\leq i,j\leq d$, the Rohlin tower over $I^{(n)}_{j}$ is obtained
stacking _subtowers_ of the Rohlin towers over $I^{(n)}$ (namely sets of the
form $\\{T^{k}J,0\leq k<q^{(n)}_{j}\\}$ for some subinterval $J\subset
I^{(n)}_{j}$). Then $(Z_{n})_{ij}$ is the number of subtowers of the Rohlin
tower over $I^{(n)}_{i}$ inside the Rohlin tower over $I^{(n+1)}_{j}$. It
follows that the Rohlin tower over $I^{(n+1)}_{j}$ is made by stacking exactly
$\sum_{i=1}^{d}(Z_{n})_{ij}$ subtowers of Rohlin towers of step $n$. Notice
that $\sum_{i=1}^{d}(Z_{n})_{ij}$ is the norm of the $j^{th}$ _column_ of the
matrix $Z_{n}$.

####  2.5.8. Length cocycle

One can check that the length (column) vectors
$(\lambda^{(k)})_{k\in\mathbb{N}}$ that give the lengths
$\lambda^{(k)}_{j}=|I^{(k)}_{j}|$ of the exchanged intervals of the induced
map $T_{k}$ on the sequence of inducing intervals
$\\{I^{(k)},k\in\mathbb{N}\\}$ given by the Zorich acceleration also transform
via a cocycle and that the cocycle is exactly the transpose inverse
$(Z^{\dagger})^{-1}$ of the Zorich cocyle $Z$. Thus, we have that

$\lambda^{(m)}=Z(m,n)^{\dagger}\lambda^{(n)}=Z(m)^{\dagger}Z(m-1)^{\dagger}\cdots
Z(n-1)^{\dagger}\lambda^{(n)},\qquad\text{for\ every}\ 0\leq m<n.$

Let us also recall that, for every cocycle product $B:=Z(0,n)=Z^{(n)}(T)$ , if
we define the sub-simplex

(13)
$\Delta_{B}:=\left\\{\lambda=\frac{B^{\dagger}\lambda}{|B^{\dagger}\lambda|},\qquad\lambda\in\Delta_{d-1}\right\\}\subset\Delta_{d}\subset\mathbb{R}^{+}_{d},$

(where $B^{\dagger}$ is as above the transpose of $B$), then (because of the
relation between lengths and Zorich cocycle), for any
$\lambda^{\prime}\in\Delta_{d}$, if $T^{\prime}$ is the IET with length data
$\lambda^{\prime}$ and same combinatorics $\pi$ than $T$, we have that
$Z^{(n)}(T^{\prime})=Z^{(n)}(T)=B$.

####  2.5.9. Cocycle action on log-slopes of AIETs

Let us now consider affine interval exchange $T\in\mathcal{A}_{d}$ that it is
infinitely renormalizable, and assume that its rotation number $\gamma(T)$ is
_irrational_. Then, Zorich acceleration is well-defined for $T$ (see 2.3.6).
An important fact is that the action of $\mathcal{Z}$ on the slopes column
vector $\rho(T)=(\rho_{1},\cdots,\rho_{d})^{\dagger}$ satisfies the following:
if
$\omega(T):=\log\rho(T)=\big{(}\log\rho_{1},\cdots,\log\rho_{d}\big{)}^{\dagger}$
denotes what we call the _log-slope vector_ of $T$ (whose entries are the
logarithms of the slopes of $T$ on continuity intervals), we have

$\omega(\mathcal{R}T)=Z(T)\,\omega(T),\qquad\omega(\mathcal{R}^{n}T)=Q(n)\omega(T),\qquad\text{for\
all\ }T\in\mathcal{A}_{d},\ n\in\mathbb{N}.$

Thus, the way log-slope vector $\omega(T)$ transform under $\mathcal{V}$ does
not depend on the value of $\lambda(T)$ and is linear and given by the Zorich
cocycle.

####  2.5.10. Lyapunov exponents and Oseledets splittings

Zorich showed in [Zo:gau] that for every irreducible
$\pi\in\mathfrak{S}^{0}_{d}$ the Zorich cocycle ${Z}:\mathcal{I}_{\pi}\to
SL(d,\mathbb{Z})$, as well as its traspose dual $(Z^{\dagger})^{-1}$, are
_integrable_ (see § 2.5.5) with respect to the Zorich measure
$\mu_{\mathcal{Z}}$ (introduced in § 2.5.1). Let us consider the natural
extension $\hat{\mathcal{Z}}:\hat{\mathcal{I}_{d}}\to\hat{\mathcal{I}_{d}}$
(see § 2.5.2) with its invariant measure ${\mu}_{\hat{\mathcal{Z}}}$. The
cocycle $Z$ over $\mathcal{Z}$ can be _extended_ to a cocycle, which we still
denote by $Z$, over $\hat{\mathcal{Z}}$, by defining
$Z:\hat{\mathcal{I}_{d}}\to SL(d,\mathcal{Z})$ to be given by
$Z(\hat{T}):=Z(p(\hat{T}))$ for every $\hat{T}\in\hat{\mathcal{I}_{d}}$. Then,
$Z$ is now a cocycle over $\mathcal{Z}$ which is still integrable (by
definition of $\hat{Z}$ and by construction of $\mu_{\hat{\mu}}$, see in
particular property $(ii)$ of the natural extension, see § 2.5.2). Notice also
that, by definition of the extension, if $\hat{T}\in p^{-1}(T)$ (since then
$Z(\hat{T})=Z(T)$), then
$Z_{n}(T)=Z(\mathcal{Z}^{n}T)=Z(\hat{\mathcal{Z}}^{n}\hat{T})=Z_{n}(\hat{T})$
for every $n\in\mathbb{N}$.

As a consequence of Oseledets theorem for cocycles over _invertible_
transformations and of the celebrated works333333The _symmetry_ of the
Lyapunov exponents (i.e. the property that for every exponent $\lambda_{i}$
also $-\lambda_{i}$ is an exponent), is a consequence of the _symplectic_
nature of the Zorich cocycle (proved in [Zo:dev], [Zo:gau]), the
_hyperbolicity_ , namely the inequality $\lambda_{g}>0$, was proved by Forni
[Fo:ann] and _simplicity_ , namely $\lambda_{i}<\lambda_{i+1}$ for every
$1\leq i<g$ was proved by Avila and Viana [AV:sim]. [Ve:Teich, Zo:dev, Fo:ann,
AV:sim] it hence follows that there exists $g$ (positive) _Lyapunov exponents_
$\lambda_{1}>\ldots>\lambda_{g}>0$ (where $g$ is the genus of the suspension
$\hat{T}$) such that for $\mu_{\mathcal{Z}}$-almost every
$\hat{T}\in\hat{\mathcal{I}}_{\pi}$, there exists a _splitting_ , namely,
denoting by $\kappa$ the number of singularities of the suspension $\hat{T}$
(see § 2.5.2 and § 2.1.6), a decomposition

(14) $\mathbb{R}^{d}=\bigoplus_{-g\leq i\leq
g}E_{i}(\hat{T}),\qquad\text{where}\ \dim
E_{i}(\hat{T})=\begin{cases}1&\text{if}\ i\neq 0,\\\ \kappa-1&\text{if}\
i=0\end{cases}$

(called _Oseledets splitting_) such that:

(15)
$\lim_{n\to\pm\infty}\frac{1}{n}\log\|Z^{(n)}(\hat{T})v\|=\begin{cases}\lambda_{i}&\text{
if }v\in E_{i}(\hat{T})\text{ and }i>0\\\ -\lambda_{i}&\text{ if }v\in
E_{i}(\hat{T})\text{ and }i<0,\\\ 0&\text{ if }v\in
E_{c}(\hat{T}).\end{cases}$

We define the _stable_ , _unstable_ and _central_ space for any
$n\in\mathbb{N}$ to be respectively

$E^{(n)}_{s}(\hat{T})=E_{s}(\mathcal{Z}^{n}\hat{T}):=\bigoplus_{-g\leq i\leq
0}E_{i}(\mathcal{Z}^{n}\hat{T})\qquad
E^{(n)}_{s}(\hat{T})=E_{u}(\mathcal{Z}^{n}\hat{T}):=\bigoplus_{0\leq i\leq
g}E_{i}(\mathcal{Z}^{n}\hat{T}),\qquad
E_{c}^{(n)}:=E_{c}(\mathcal{Z}^{n}\hat{T}).$

Then invariance of the splitting means that for any $m<n$ we have that

$Z(m,n)E^{(m)}_{\nu}(\mathcal{Z})=E^{(n)}_{\nu}(\mathcal{Z}),\qquad\text{where}\
\nu\ \text{in\ any\ index\ in}\ \\{u,s,c\\}.$

Furthermore, Oseledets theorem also guarantees a control of the _angle_
$\angle$ (see e.g. Section 4 in [ViaLP]) between stable, unstable and central
spaces (where the angle $\angle(V,W)$ between two linear subspaces
$V,W\subset{\mathbb{R}^{d}}$ is defined as the minimum angle $\angle(v,w)$
among all non-zero vectors $v\in V,w\in W$), namely for every $\epsilon>0$
there exists $c=c(\epsilon,\hat{T})>0$ such that

(16)
$\displaystyle\lim_{n\to\pm\infty}\frac{\sin|\angle(E^{(n)}_{\nu_{1}}(\hat{T}),E^{(n)}_{\nu_{2}}(\hat{T}))|}{n}=0\qquad\text{for\
all\ \emph{distinct}\ pair\ of\ indexes}\ \nu_{1},\nu_{2}\in\\{u,s,c\\}.$

We say that a IET is _Oseledets generic_ if it satisfies all the conclusions
of Oseledets theorem listed above. An application of the Oseledets theorem (or
simply ergodicity combined with integrability) also shows that if $T$ is
Oseledets generic, then

(17) $\lim_{n\to\pm\infty}\frac{1}{n}\log\|Z_{n}(\hat{T})\|=0.$

[Notice that here we consider the elementary matrix
$Z_{n}(\hat{T})=Z(\hat{\mathcal{Z}}^{n}\hat{T})$ and not $Z^{(n)}(\hat{T})$
which is a _product_ of cocycle matrices.]

###  2.6. Birkhoff sums and special Birkhoff sums

The Zorich cocycle plays a crucial role also in the study of Birkhoff sums of
functions over an IET, through the study of _special Birkhoff sums_ , which
were introduced in the work by Marmi-Moussa-Yoccoz [MMY] as fundamental blocks
to decompose and study Birkhoff sums over an IET. We recall here both the
definition of Birkhoff sums and special Birkhoff sums, as well as the
connection between special Birkhoff sums and the (extended) Zorich cocycle,
while in the next subsection § 2.6.2 we recall how special Birkhoff sums can
be used to decompose Birkhoff sums.

####  2.6.1. Piecewise continuous functions and Birkhoff sums

Given a GIET $T$, let us denote (using a notation inspired by [MMY])
$\mathcal{C}(T):=\mathcal{C}\left(\sqcup_{j=1}^{d}I^{t}_{j}\right)$ (or for
short $\mathcal{C}\left(\sqcup I^{t}_{j}\right)$, the space of _piecewise
continuous_ functions $f:[0,1]\to\mathbb{R}$ which are _continuous_ on each
continuity interval $I^{t}_{j}$, for $1\leq j\leq d$ of $T$ and extend to
continuous functions on the closure of each $I^{t}_{j}$ (so that right and
left limit as $x$ tend to the endpoints of each $I^{t}_{\alpha}$ exist).
Notice that as subspace, it contains the space
$\Gamma=\Gamma^{(0)}:=\Gamma\left(\sqcup_{j=1}^{d}I^{t}_{j}\right)$ of
functions which are _piecewise constant_ on each interval $I^{t}_{j}$, $1\leq
j\leq d$.

Given $\mathcal{C}\left(\sqcup_{j=1}^{d}I^{t}_{j}\right)$, the _Birkhoff sum_
$S_{n}f(x)$ of $f$ over $T$ denotes the sum of $f$ along the orbit up of $x$
under $T$ to time $n$, i.e. $S_{n}f(x):=\sum_{j=1}^{n-1}f(T^{j}x).$ We stress
that in this paper we consider in general Birkhoff sums over a _generalized_
IET, while in [MMY] and the consequent works one usually restricts the
attention to Birkhoff sums over _standard_ IETs.

####  2.6.2. Special Birkhoff sums

Assume that $T$ is infinitely renormalizable and let
$\\{I^{(n)},n\in\mathbb{N}\\}$, where $I^{(n)}=[0,a_{n}]$, be a sequence of
intervals obtained by the Zorich (or more in general by a further)
acceleration of Rauzy-Veech induction and let $\\{T_{n},n\in\mathbb{N}\\}$ be
the corresponding induced GIETs, $T_{n}$ being the first return of $T$ on
$[0,a_{n}]$. Let $q^{(n)}={(q^{(n)}_{1},\dots,q^{(n)}_{d})}^{\dagger}$ be the
corresponding vector of first return times, or equivalently, of _heights_ of