David Ralston, Serge Troubetzkoy
We show that for odd-valued piecewise-constant skew products over a certain two parameter family of interval exchanges, the skew product is ergodic for a full-measure choice of parameters.
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Ergodicity of explicit logarithmic cocycles over IETs
本文证明了区间交换变换上具有对数奇异性的余循环所构造的斜积扩张的遍历性，并给出了亏格二中极小局部哈密顿流的显式遍历扩张例子。
The main subject of the paper, motivated by a question raised by Boshernitzan, is to give criteria for a bounded complex-valued sequence to be uncorrelated to any strictly ergodic sequence. As a tool developed to study this problem we introduce the notion of correlation between two shift-invariant measures supported by the symbolic space with complex symbols. We also prove a "lifting lemma" for generic points: given a joining $ξ$ of two shift-invariant measures $μ$ and $ν$, every point $x$ generic for $μ$ lifts to a pair $(x,y)$ generic for $ξ$ (such $y$ exists in the full symbolic space). Thi…
Ergodicity of cocyles over 2-dimensional rotations
本文研究二维旋转上余循环的回复性与遍历性，考虑坏逼近无理数旋转及满足丢番图条件的不连续函数，主要关注ρ=2，d=1和2的情形。
Let T_αdenote the rotation T_αx=x+α(mod 1) by an irrational number αon the additive circle T=[0,1). Let β_1,..., β_d be d\geqslant 1 parameters in [0, 1). One of the goals of this paper is to describe the ergodic properties of the cocycle (taking values in R^(d+1)) generated over T_αby the vectorial function Ψ_{d+1}(x):=(ϕ(x), ϕ(x+β_1),..., ϕ(x+β_d)), with ϕ(x)={x}-1/2. It was already proved in \cite{LeMeNa03} that Ψ_{2} is regular for αwith bounded partial quotients. In the present paper we show that Ψ_{2} is regular for any irrational α. For higher dimensions, we give sufficient conditions f…
For $θ\in [0,1]$, we consider the map $T_\a: \T^2 \to \T^2$ given by $T_θ(x,y)=(x+θ,y+2x+θ)$. The skew product $f_\a: \T^2 \times \C \to \T^2 \times \C$ given by $f_θ(x,y,z)=(T_θ(x,y),z+e^{2 πi y})$ generates the so called Weyl sums cocycle $a_\a(x,n) = \sum_{k=0}^{n-1} e^{2πi(k^2θ+kx)}$ since the $n^{\rm th}$ iterate of $f_\a$ writes as $f_\a^n(x,y,z)=(T_\a^n(x,y),z+e^{2πiy} a_\a(2x,n))$. In this note, we improve the study developed by Forrest in \cite{forrest2,forrest} around the density for $x \in \T$ of the complex sequence ${\{a_\a(x,n)\}}_{n\in \N}$, by proving the ergodicity of $f_θ$ fo…
We consider interval exchange transformations of periodic type and construct different classes of recurrent ergodic cocycles of dimension $\geq 1$ over this special class of IETs. Then using Poincaré sections we apply this construction to obtain recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.
We show that under some natural ergodicity assumptions extensions given by Rokhlin cocycles lift the multiplier property if the associated locally compact group extension has only countably many L^\infty-eigenvalues. We make use of some analogs of basic results from the theory of finite-rank modules associated to an extension of measure-preserving systems in the setting of a non-singular base.
On the ergodicity of anti-symmetric skew products with singularities and its applications
本文提出一种新方法证明具有端点奇异性的区间交换变换斜积的遍历性，适用于非对称协环，改进了传统对数奇异性的限制，并应用于局部哈密顿流谱分解的误差项等分布研究。
A criterion for unique ergodicity for points of a curve in the space of interval exchange transformation is given.
The central limit theorem for Markov chains generated by iterated function systems consisting of orientation preserving homeomorphisms of the interval is proved. We study also ergodicity of such systems.
Effective intrinsic ergodicity for expanding interval maps
将Einsidler等人关于内在遍历性的结果推广到扩张映射（包括β-变换）的简单情形，证明了有效内在遍历性。
Nondegeneracy of the spectrum of the twisted cocycle for interval exchange transformations
本文证明了与区间交换变换相关的扭曲谱上循环顶部 Lyapunov 指数为正，通过联系亏格趋于无穷的仿射不变子流形族的指数极限，并应用于偏差估计及谱测度局部维数公式。
We give conditions for minimality of $\mathbb Z/N\mathbb Z$ extensions of a rotation of angle $α$ with one marked point, solving the problem for any prime $N$: for $N=2$, these correspond to the Veech 1969 examples, for which a necessary and sufficient condition was not known yet. We provide also a word combinatorial criterion of minimality valid for general interval exchange transformations, which applies to $\mathbb Z/N\mathbb Z$ extensions of any interval exchange transformation with any number of marked points. Then we give a condition for unique ergodicity of these extensions when the ini…
We study the existence of transitive exchange maps with flips defined on the unit circle. We provide a complete answer to the question of whether there exists a transitive exchange map of the unit circle defined on n subintervals and having f flips.
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.
Exchange of Intervals and Intrinsic Ergodicity of the Negative Beta shift
研究负贝塔移位的内在ergodic性，探讨代数整数序列与最大熵测度的关系，揭示黄金比例与贝塔值的关联。
Unique ergodicity for random noninvertible maps on an interval
本文研究单位区间上不可逆随机动力系统，给出平均单射系统唯一遍历性的简便条件，并构造反例说明条件失效情形。
Twisted cocycle for interval exchange transformations: Invariant structures and Lyapunov spectrum
本文研究了区间交换变换的扭曲上循环，通过不变子丛分解和不变辛形式证明了零 Lyapunov 指数的存在性与谱对称性，并揭示了旋转型置换的谱退化现象。
We consider smooth area-preserving flows (also known as locally Hamiltonian flows) on surfaces of genus $g\geq 1$ and study ergodic integrals of smooth observables along the flow trajectories. We show that these integrals display a \emph{power deviation spectrum} and describe the cocycles that lead the pure power behaviour, giving a new proof of results by Forni (Annals 2002) and Bufetov (Annals 2014) and generalizing them to observables which are non-zero at fixed points. This in particular completes the proof of the original formulation of the Kontsevitch-Zorich conjecture. Our proof is base…
On the ergodicity of infinite antisymmetric extensions of symmetric IETs
该研究针对对称区间交换变换上的无限反对称斜积扩张，证明在几乎所有对称组合数据下，由分段指示函数上循环定义的 $\mathbb Z$ 值斜积关于 Lebesgue–计数乘积测度遍历。
Proofs of ergodicity of piecewise Möbius interval maps using planar extensions
本文通过平面扩张和“有界非满值域”等新条件，结合“绗缝”操作，证明了分段莫比乌斯区间映射的遍历性等动力学性质，并成功应用于Nakada α-连分数及一类Fuchsian群区间映射。
Unique ergodicity for noninvertible function systems on an interval
本文研究单位区间上连续函数构成的随机动力系统，借助有界变差给出其唯一遍历性的充分条件，并辅以多类示例说明。
After relating the notion of $ω$-recurrence in skew products to the range of values taken by partial ergodic sums and Lyapunov exponents, ergodic $\mathbb{Z}$-valued cocycles over an irrational rotation are presented in detail. First, the generic situation is studied and shown to be $1/n$-recurrent. It is then shown that for any $ω(n) <n^{-ε}$, where $ε>1/2$, there are uncountably many infinite staircases (a certain specific cocycle over a rotation) which are \textit{not} $ω$-recurrent, and therefore have positive Lyapunov exponent. A further section makes brief remarks regarding cocycles over…
Non-commutative skew-product extension dynamical systems
本文研究了交叉积上的非交换斜积扩张动力系统，证明了扩张共轭与上闭链上同调的等价性，并刻画了唯一遍历性及不变态集合。
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 give sufficient conditions for intervals $(a,b)$ such that the associated open dynamical system for the doubling map is intrinsically ergodic. We also show that the set of parameters $(a,b) \in (\frac{1}{4}, \frac{1}{2}) \times (\frac{1}{2},\frac{3}{4})$ such that the attractor $(Λ_{(a,b)}, f_{(a,b)})$ is intrinsically ergodic has full Lebesgue measure and we construct a set of points where intrinsic ergodicity does not hold.
Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with non-specular reflections laws. We study the absolutely continuous invariant probabilities of the slap maps for several polygons, including regular polygons and triangles. We also present a general method for constructing polygons with slap maps having more than one ergodic absolutely continuous invariant probability.
Coexistence of two contrasting recurrence properties of certain non-integrable cocycles
本文研究了非积分协振子的递归性质，证明了此类系统在耗散的同时具有拓扑递归性。
Ergodic properties of infinite extension of symmetric interval exchange transformations
本文研究了对称区间交换变换的无限扩张的遍历性质，证明了特定余循环的斜积在遍历基下是遍历的，并在唯一遍历性假设下推广到更一般的函数，最后通过弱混合情形得到无限遍历指数。
Let $ β$ be a real number less than -1. In this paper, we prove the uniqueness of the measure with maximal entropy of the negative $β$-shift. Endowed with the shift, this symbolic dynamical system is coded under certain conditions, but in all cases, it is shown that the measure with maximal entropy is carried by a support coded by a recurrent positive code. One of the difference between the positive and the negative $β$-shift is the existence of gaps in the system for certain negative values of $ β$ . These are intervals of negative $β$-representations (cylinders) negligible with respect to th…
Ergodicity of skew-products over typical IETs
该研究针对一类由区间交换变换与有限间断分段常值函数构造的斜积无穷测度保持系统，证明了在参数典型选取下，其关于\\( {Leb}_{[0,1)\times \mathbb{R}} \\)测度具有遍历性，拓展了无穷测度系统遍历性的研究结果。
Exponential ergodicity in the bounded-Lipschitz distance for a subclass of piecewise-deterministic Markov processes with random switching between flows
本文研究了一类在流之间随机切换的分段确定性马尔可夫过程，提出了仅依赖流性质和跳跃后位置转移律的可验证条件，以保证其在有界Lipschitz距离下的指数遍历性，并为适用于生物模型的特例建立了简单判据。
Skew-product systems over infinite interval exchange transformations
本文研究以旋转里程计为底的无限区间交换变换 ℤ-扩张斜积系统，分析其递归、偏差、扩散与遍历性，并关联无限亏格阶梯流形上的平行流。
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.
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…
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…
Joint ergodicity of piecewise monotone interval maps
本文研究分段单调区间映射的一致联合遍历性，建立方便判据并给出区间交换变换、β-变换与Gauss映射的联合遍历应用，同时证明斜帐篷映射和单位圆上有限Blaschke乘积限制的联合混合性。
We consider the regularity of measurable solutions $χ$ to the cohomological equation \[ ϕ= χ\circ T -χ, \] where $(T,X,μ)$ is a dynamical system and $ϕ\colon X\rightarrow \R$ is a $C^k$ valued cocycle in the setting in which $T \colon X\rightarrow X$ is a piecewise $C^k$ Gibbs--Markov map, an affine $β$-transformation of the unit interval or more generally a piecewise $C^{k}$ uniformly expanding map of an interval. We show that under mild assumptions, bounded solutions $χ$ possess $C^k$ versions. In particular we show that if $(T,X,μ)$ is a $β$-transformation then $χ$ has a $C^k$ version, thus…
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_+…
Ergodic measures for periodic type $\mathbb{Z}^m$-skew-products over Interval Exchange Transformations
本文研究了区间交换变换上周期型$\mathbb{Z}^m$-斜积的遍历不变Radon测度分类问题，通过引入adic编码的符号语言并应用相关符号结果，证明了此类测度恰为由$\mathbb{R}^m$参数化的Maharam测度，并给出了其显式形式与弱*-连续性。
It is known since 40 years old paper by M. Keane that minimality is a generic (i.e. holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" property. We conjecture in this paper that this occurs if and only if the linear restrictions contain a Lagrangian subspace of the first homology of the suspension surface. We partially prove it in the "only if" direction and provide a series of examples to support the converse on…
On the uniqueness of affine IETs semi-conjugated to IETs
本文证明了几乎每个不可约区间交换变换$T$，在其中心稳定空间中任意向量$ω$下，存在唯一的归一化AIET半共轭于$T$且对数斜率向量为$ω$，部分回答了Marmi等提出的问题。
Isomorphisms and slow entropy of deterministic $[T,T^{-1}]$ systems
本文研究确定性$[T,T^{-1}]$系统的同构刚性，证明斜积同构条件，并构造任意低慢熵及违反慢熵变分原理的新例子。
A stretched-exponential bound is obtained for the decay of correlations of the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations. A Corollary is the Central Limit Theorem for the Teichmueller flow on the moduli space of abelian differentials. The proof involves an approximation of the induction map by a Markov chain satisfying the Doeblin condition, the method of Sinai and Bunimovich--Sinai. The main estimate is Lemma 4. As a corollary of such "loss of memory" estimates, integrability of a small power of the Kontsevich-Zorich cocycle is obtained, thus yielding …
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