Interval Exchange Research Articles

Renormalizing an infinite rational IET

We study an interval exchange transformation of \begin{document}$ [0, 1] $\end{document} formed by cutting the interval at the points \begin{document}$ \frac{1}{n} $\end{document} and reversing the order of the intervals. We find that the transformation is periodic away from a Cantor set of Hausdorff dimension zero. On the Cantor set, the dynamics are nearly conjugate to the \begin{document}$ 2 $\end{document} –adic odometer.

On Roth type conditions, duality and central Birkhoff sums for i.e.m.

We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the \emph{dual Roth type} condition is a condition on the \emph{backward} rotation number of a translation surface. We show that results on the cohomological equation previously proved in \cite{MY} for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted \emph{absolute} Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with \emph{subpolynomial} deviations of ergodic averages (corresponding to relative homology classes) \emph{distributional} limit shapes, which are constructed in a similar way to the \emph{limit shapes} of Birkhoff sums associated in \cite{MMY3} to functions which correspond to positive Lyapunov exponents.

On Roth type conditions, duality and central Birkhoff sums for i.e.m.

We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the \emph{dual Roth type} condition is a condition on the \emph{backward} rotation number of a translation surface. We show that results on the cohomological equation previously proved in \cite{MY} for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted \emph{absolute} Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with \emph{subpolynomial} deviations of ergodic averages (corresponding to relative homology classes) \emph{distributional} limit shapes, which are constructed in a similar way to the \emph{limit shapes} of Birkhoff sums associated in \cite{MMY3} to functions which correspond to positive Lyapunov exponents.

Invariant measures for interval translations and some other piecewise continuous maps

We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation maps) a Borel probability non-atomic invariant measure exists for any map. We use this result to demonstrate that any interval translation map endowed with such a measure is metrically equivalent to an interval exchange map. Finally, we study the general case of piecewise continuous maps and prove a simple result on existence of an invariant measure provided all discontinuity points are wandering.

A switched server system semiconjugate to a minimal interval exchange

Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behaviour: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewise contractions and interval exchange transformations (IETs) with flips. The ultimate result is a switched server system whose Poincaré map is semiconjugate to a minimal and uniquely ergodic IET with flips.

Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces

We consider typical area-preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affirmatively Rokhlin’s multiple mixing question in this context. The main tool is a variation of the Ratner property originally proved by Ratner for the horocycle flow, i.e. the switchable Ratner property introduced by Fayad and Kanigowski for special flows over rotations. This property, which is of independent interest, provides a quantitative description of the parabolic behavior of these flows and has implications for joining classification. The main result is formulated in the language of special flows over interval exchange transformations with asymmetric logarithmic singularities. We also prove a strengthening of one of Fayad and Kanigowski’s main results, by showing that Arnold’s flows are mixing of all orders for almost every location of the singularities.

Centralizers in the group of interval exchange transformations

We study the group of interval exchange transformations. Let T be an m-interval exchange transformation. By the rank of T we mean the dimension of the ℚ-vector space spanned by the lengths of the exchanged subintervals. We prove that if T satisfies Keane’s infinite distinct orbit condition and rank(T) > 1 + [m/2], then the only interval exchange transformations which commute with T are its powers. In the case that T is a minimal 3-interval exchange transformation, we prove a more precise result: T has a trivial centralizer in the group of interval exchange transformations if and only if T satisfies the infinite distinct orbit condition.

Derived sequences of complementary symmetric Rote sequences

Complementary symmetric Rote sequences are binary sequences which have factor complexityC(n) = 2nfor all integersn≥ 1 and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derived sequences to the prefixes of any complementary symmetric Rote sequencevwhich is associated with a standard Sturmian sequenceu. We show that any non-empty prefix ofvhas three return words. We prove that any derived sequence ofvis coding of three interval exchange transformation and we determine the parameters of this transformation. We also prove thatvis primitive substitutive if and only ifuis primitive substitutive. Moreover, if the sequenceuis a fixed point of a primitive morphism, then all derived sequences ofvare also fixed by primitive morphisms. In that case we provide an algorithm for finding these fixing morphisms.

Triangle Tiling Billiards and the Exceptional Family of their Escaping Trajectories: Circumcenters and Rauzy Gasket

Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighboring tiles in such a way that the coefficient of refraction is equal to –1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set We also give a precise description of the exceptional family of trajectories (of zero measure): these trajectories escape nonlinearly to infinity and approach fractal-like sets. We show that this exceptional family is parametrized by the famous Rauzy gasket. This proves several conjectures stated previously on triangle tiling billiards. In this work, we also give a more precise understanding of fully flipped minimal exchange transformations on 3 and 4 intervals by proving that they belong to a special hypersurface. Our proofs are based on the study of Rauzy graphs for interval exchange transformations with flips.

Spatial Limit Theorem for Interval Exchange Transformations

We prove a spatial limit theorem for generic interval exchange transformations (IETs): for a generic IET the normalized ergodic sums of a sufficiently regular (e.g., Lipschitz) function have the same asymptotic behavior of distributions as the behaviour of ergodic integrals for a generic translation flow on a flat surface, described by A. Bufetov.

Distortion in groups of affine interval exchange transformations

In this paper, we study distortion in the group \mathcal A of affine interval exchange transformations (AIET). We prove that any distorted element f of \mathcal A has an iterate f^k that is conjugate by an element of \mathcal A to a product of infinite order restricted rotations, with pairwise disjoint supports. As consequences, we prove that no Baumslag–Solitar group, BS (m,n) with \vert m \vert \neq \vert n \vert , acts faithfully by elements of \mathcal A ; every finitely generated nilpotent group of \mathcal A is virtually abelian and there is no distortion element in \mathcal A_{\mathbb Q} , the subgroup of \mathcal A consisting of rational AIETs.

A Sufficient Condition for Absolute Continuity of Conjugations between Interval Exchange Maps

A class of topological equivalent generalized interval exchange maps of genus one and of the same bounded combinatorics is considered in the paper. A sufficient condition for absolute continuity of the conjugation between two maps from this class is provided

Quantitative shrinking target properties for rotations and interval exchanges

This paper presents quantitative shrinking target results for rotations and interval exchange transformations. To do this a quantitative version of a unique ergodicity criterion of Boshernitzan is established.

Ergodicity of skew products over linearly recurrent IETs

We prove that the skew product over a linearly recurrent interval exchange transformation defined by almost any real-valued, mean-zero linear combination of characteristic functions of intervals is ergodic with respect to Lebesgue measure.

Cascades in the dynamics of affine interval exchange transformations

We describe in this article the dynamics of a one-parameter family of affine interval exchange transformations. This amounts to studying the directional foliations of a particular dilatation surface introduced in Duryev et al [Affine surfaces and their Veech groups. Preprint, 2016, arXiv:1609.02130], 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. This study is achieved through analysis of the dynamics of the Veech group of this surface combined with a modified version of Rauzy induction in the context of affine interval exchange transformations.

Cohomological equation and local conjugacy class of Diophantine interval exchange maps

We extend some results of Marmi--Moussa--Yoccoz on the cohomological equations and local conjugacy classes of interval exchange maps of restricted Roth type. In particular, we answer a question of Krikorian about the codimension of the local conjugacy class of self-similar interval exchange maps associated to the Eierlegende Wollmilchsau and the Ornithorynque.

Rigidity of square-tiled interval exchange transformations

We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction \begin{document}$ \theta $\end{document} on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if \begin{document}$ \tan\theta $\end{document} has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and \begin{document}$ \tan\theta $\end{document} has bounded partial quotients, the square-tiled interval exchange transformation \begin{document}$ T $\end{document} is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations.

Möbius disjointness for interval exchange transformations on three intervals

We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.

Full families of generalized interval exchange transformations

We consider generalized interval exchange transformations, or briefly GIETs, that is bijections of the interval which are piecewise increasing homeomorphisms with finitely many branches. When all continuous branches are translations, such maps are classical interval exchange transformations, or briefly IETs. The well-known Rauzy renormalization procedure extends to a given GIET and a Rauzy renormalization path is defined, provided that the map is infinitely renormalizable. We define full families of GIETs, that is optimal finite dimensional parameter families of GIETs such that any prescribed Rauzy renormalization path is realized by some map in the family. In particular, a GIET and a IET with the same Rauzy renormalization path are semi-conjugated. This extends a classical result of Poincaré relating circle homeomorphisms and irrational rotations.

Balancedness and coboundaries in symbolic systems

This paper studies balancedness for infinite words and subshifts, both for letters and factors. Balancedness is a measure of disorder that amounts to strong convergence properties for frequencies. It measures the difference between the numbers of occurrences of a given word in factors of the same length. We focus on two families of words, namely dendric words and words generated by substitutions. The family of dendric words includes Sturmian and Arnoux-Rauzy words, as well as codings of regular interval exchanges. We prove that dendric words are balanced on letters if and only if they are balanced on words. In the substitutive case, we stress the role played by the existence of coboundaries taking rational values and show simple criteria when frequencies take rational values for exhibiting imbalancedness.