H. Bruin, S. Troubetzkoy
We study the dynamics of a class of interval translation map on three intervals. We show that in this class the typical ITM is of finite type (reduce to an interval exchange transformation) and that the complement contains a Cantor set. We relate our maps to substitution subshifts. Results on Hausdorff dimension of the attractor and on unique ergodicity are obtained.
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Typical Weak Mixing and Exceptional Spectral Properties for Interval Translation Mappings
本文研究了区间平移映射的弱混合性，证明了典型Bruin-Troubetzkoy映射的弱混合性，并扩展到其他类型映射，同时构造了非弱混合的无限类型Bruin-Troubetzkoy映射例子。
We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear~naturally.
We show that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a surface of genus $g \geq 2$ (with prescribed singularity types) is weakly mixing.
Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite type contains an open, dense, and full Lebesgue measure subset of the space of ITMs of three intervals. For this, we show that any ITM of three intervals can be reduced either to a rotation or to a double rotation.
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 …
Characterisation of Stability for Interval Translation Maps
本文研究了区间翻译映射的稳定性，提出并证明了稳定性与'无临界连接'和'匹配'两个动态性质的关系，为一般ITM的稳定性理论奠定基础。
Topological Prevalence of Finite Type Interval Translation Maps
本文研究了有限类型区间翻译映射的拓扑性质，证明对于每个r≥2，有限类型映射在所有可能的区间翻译映射中构成开密集集，验证了Boshernitzan和Kornfeld的长期猜想。
Countable IET Models and Defect Sets for Interval Translation Maps
本文针对区间平移映射，结合熵证明其几乎处处可逆，利用Arnoux-Ornstein-Weiss定理构造可数IET模型，通过缺陷集刻画有限/可数IET模型的几何特征，并显式给出Bruin-Troubetzkoy映射的可数IET模型。
We consider piecewise monotone maps, we show that an ergodic measure for which the map is invertible almost everywhere can not be mixing. It follows that every ergodic measure for an interval translation mapping is not mixing. We also show that double rotations without periodic points have an ergodic but not weakly mixing invariant measure. This article is dedicated to the memory of Anatoly Mikhailovich Stepin.
Transversality for Interval Translation Maps
本文研究了区间翻译映射的动力学子空间横截性定理，为系统扰动控制提供了理论支持，并在稳定性刻画和拓扑版本的Boshernitzan-Kornfeld猜想证明中起到关键作用。
Let $d >1$. In this paper we show that for an irreducible permutation $π$ which is not a rotation, the set of $[λ]\in \mathbb{P}_+^{d-1}$ such that the interval exchange transformation $f([λ],π)$ is not weakly mixing does not have full Hausdorff dimension. We also obtain an analogous statement for translation flows. In particular, it strengthens the result of almost sure weak mixing proved by G. Forni and the first author. We adapt here the probabilistic argument developed in their paper in order to get some large deviation results. We then show how the latter can be converted into estimates o…
Density of Stable Interval Translation Maps
本文研究区间平移映射，证明在固定分支数的参数空间中，有限型、返回映射为圆周旋转且幸存集 Hausdorff 连续变化的稳定映射构成开稠密集。
Interval Translation Maps with Weakly Mixing Attractors
针对具有 Cantor 吸引子的双参数区间平移映射族，本文利用类似 Rauzy 归纳的参数空间动力系统 $G$，判定其线性递归性或弱混合性。
In this note we show that the transfer operator of a Rauzy-Veech-Zorich renormalization map acting on a space of quasi-Hölder functions is quasicompact and derive certain statistical recurrence properties for this map and its associated Teichmüller flow. We establish Borel-Cantelli lemmas, Extreme Value statistics and return time statistics for the map and flow. Previous results have established quasicompactness in Hölder or analytic function spaces, for example the work of M. Pollicott and T. Morita. The quasi-Hölder function space is particularly useful for investigating return time statisti…
Piecewise Translations is a class of dynamical systems which arises from some applications in computer science, machine learning, and electrical engineering. In dimension 1 it can also be viewed as a non-invertible generalization of Interval Exchange Transformations. These dynamical systems still possess some features of Interval Exchanges but the total volume is no longer preserved and allowed to decay. Every Piecewise Translation has a well-defined attracting subset which is the locus of our interest. We prove some results about how fast the dynamics lock onto the attractor, geometry of the …
A translation surface on (S, Σ) gives rise to two transverse measured foliations \FF, \GG on S with singularities in Σ, and by integration, to a pair of cohomology classes [\FF], \, [\GG] \in H^1(S, Σ; \R). Given a measured foliation \FF, we characterize the set of cohomology classes \B for which there is a measured foliation \GG as above with \B = [\GG]. This extends previous results of Thurston and Sullivan. We apply this to two problems: unique ergodicity of interval exchanges and flows on the moduli space of translation surfaces. For a fixed permutation σ\in \mathcal{S}_d, the space \R^d_+…
We will establish several arithmetic and geometric properties regarding the bi-sequences of approximation coefficients (BAC) associated with the two one-parameter families of piecewise-continuous Mobius transformations introduced by Haas and Molnar. The Gauss and Renyi maps, which lead to the expansions of irrational numbers on the interval as regular and backwards continued fractions, are realized as special cases. The results are natural generalizations of theorems from Diophantine approximation.
We present an invariant density for the finite Gauss transformation of the unit interval and discuss some properties of this transformation.
This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school "Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory" (from 4 to 16 July, 2011). In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teichmüller flow and the SL(2,R)-action on these moduli spaces and the Kontsevich-Zorich cocycle over the Teichmüller geodesic flow. We sketch two applications of the ergodic properties of the Teichmüller flo…
We propose an extension of ergodic theory which focuses on the identification of ergodicity in terms of the uniqueness of the invariant measure. We first explain the concept for the doubling maps, which can be analyzed using Fourier methods. We then proceed to the Gauss-type maps of interest, of the form $x\mapsto -β/x$ mod $2\mathbb Z$ on the symmetric interval $[-1,1]$, for $0<β\le1$. We study an extended state space on the interval, formed as the restriction to the interval $[-1,1]$ of functions of the form $f+\mathbf{H}g$, where $f$ and $g$ are $L^1$-functions. We then look for invariant s…
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…
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