Affine interval exchange transformations with flips and wandering intervals

There exist uniquely ergodic affine interval exchange transformations of [0,1] with flips having wandering intervals and such that the support of the invariant measure is a Cantor set.Comment: 7 page

Cascades in the dynamics of affine interval exchange transformations

We describe in this article the dynamics of a $1$-parameter family of affine interval exchange transformations. It amounts to studying the directional foliations of a particular affine surface, the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial, but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. s study is achieved through the analysis the dynamics of the Veech group of this surface combined a modified version of Rauzy induction in the context of affine interval exchange transformations

Affine interval exchange maps with a wandering interval

For almost all interval exchange maps T_0, with combinatorics of genus g>=2, we construct affine interval exchange maps T which are semi-conjugate to T_0 and have a wandering interval.Comment: 43 pages, 1 figur

Linearization of generalized interval exchange maps

A standard interval exchange map is a one-to-one map of the interval which is locally a translation except at finitely many singularities. We define for such maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine arithmetical condition called restricted Roth type which is almost surely satisfied in parameter space. Let $T_0$ be a standard interval exchange map of restricted Roth type, and let $r$ be an integer $\geq 2$. We prove that, amongst $C^{r+3}$ deformations of $T_0$ which are $C^{r+3}$ tangent to $T_0$ at the singularities, those which are conjugated to $T_0$ by a $C^r$ diffeomorphism close to the identity form a $C^1$ submanifold of codimension $(g-1)(2r+1) +s$. Here, $g$ is the genus and $s$ is the number of marked points of the translation surface obtained by suspension of $T_0$. Both $g$ and $s$ can be computed from the combinatorics of $T_0$.Comment: 52 pages. This version includes a new section where we explain how to adapt our result to the setting of perturbations of linear flows on translation surface

DEVIATION OF ERGODIC AVERAGES FOR SUBSTITUTION DYNAMICAL SYSTEMS WITH EIGENVALUES OF MODULUS ONE

International audienceDeviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of mod-ulus 1. The functions γ we consider are the corresponding eigen-functions. In Theorem 1.1 we prove that the limit inferior of the ergodic sums (n, γ(x 0) +. .. + γ(x n−1)) n∈N is bounded for every point x in the phase space. In Theorem 1.2, we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length. Under additional assumptions, we prove that ergodic integrals satisfy the Central Limit Theorem (Theorem 1.3, Theorem 1.9)

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

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 C1 -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≥ 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 in ly renormalizable, irrational GIETs of any number of intervals d≥ 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