Aperiodic Lorentz gas: recurrence and ergodicity

Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces

We discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular we show that a closed subgroup of a product of finitely many linear groups over local fields supports a recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces. The structure of closed subgroups of linear groups over local fields and the properties of group actions with respect to stationary me…

非周期系统中与划分无关的Alpern引理

Alpern Lemma Independent of a Partition for Aperiodic Systems

该研究证明非周期保概率系统中存在基与有限可测划分无关的Alpern多塔，消除了相关引理及极限定理结果的遍历性假设，拓展了其适用范围。

非周期镶嵌上洛伦兹气体的统计性质，第一部分

Statistical properties of Lorentz gases on aperiodic tilings, Part 1

本文研究有限局部复杂性非周期重复镶嵌上无角点且有限视界的洛伦兹气体模型，证明碰撞映射紧因子具有K性质，并由此得到模式等变函数的混合性及气体流的平面遍历性。

Recurrence quantification analysis of the period-doubling sequence

The period-doubling sequence is one of the most well-known aperiodic $0$-$1$ sequences. In this paper, a complete description of its symbolic recurrence plot is given, and formulas for asymptotic values of basic recurrence quantifiers are derived.

On multiple recurrence and other properties of "nice" infinite measure preserving transformations

We discuss multiple versions of rational ergodicity and rational weak mixing for "nice" transformations, including Markov shifts, certain interval maps and hyperbolic geodesic flows. These properties entail multiple recurrence.

Relation between powers of factors and recurrence function characterizing Sturmian words

In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved by Carpi and de Luca, and Damanik and Lenz.

二维旋转上的余循环的遍历性

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。

Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems

The Furstenberg recurrence theorem (or equivalently, Szemerédi's theorem) can be formulated in the language of von Neumann algebras as follows: given an integer $k \geq 2$, an abelian finite von Neumann algebra $(\M,τ)$ with an automorphism $α: \M \to \M$, and a non-negative $a \in \M$ with $τ(a)>0$, one has $\liminf_{N \to \infty} \frac{1}{N} \sum_{n=1}^N \Re τ(a α^n (a) ... α^{(k-1)n} (a)) > 0$; a subsequent result of Host and Kra shows that this limit exists. In particular, $\Re τ(a α^n (a) >... α^{(k-1)n} (a)) > 0$ for all $n$ in a set of positive density. From the von Neumann algebra pers…

Effective intrinsic ergodicity for countable state Markov shifts

For strongly positively recurrent countable state Markov shifts, we bound the distance between an invariant measure and the measure of maximal entropy in terms of the difference of their entropies. This extends an earlier result for subshifts of finite type, due to Kadyrov. We provide a similar bound for equilibrium measures of strongly positively recurrent potentials, in terms of the pressure difference. For measures with nearly maximal entropy, we have new, and sharp, bounds. The strong positive recurrence condition is necessary.

关于全纯对应的诱导系统与遍历性

On induced systems and ergodicity for holomorphic correspondences

本文针对全纯对应建立庞加莱重现与卡茨定理，引入诱导对应并证明其继承遍历性，同时刻画遍历测度、证明前向轨道稠密性及罗赫林-角谷型引理。

Infinite-horizon Lorentz tubes and gases: recurrence and ergodic properties

We construct classes of two-dimensional aperiodic Lorentz systems that have infinite horizon and are 'chaotic', in the sense that they are (Poincaré) recurrent, uniformly hyperbolic, ergodic, and the first-return map to any scatterer is $K$-mixing. In the case of the Lorentz tubes (i.e., Lorentz gases in a strip), we define general measured families of systems (\emph{ensembles}) for which the above properties occur with probability 1. In the case of the Lorentz gases in the plane, we define families, endowed with a natural metric, within which the set of all chaotic dynamical systems is uncoun…

On the complexity function for sequences which are not uniformly recurrent

We prove that every non-minimal transitive subshift $X$ satisfying a mild aperiodicity condition satisfies $\limsup c_n(X) - 1.5n = \infty$, and give a class of examples which shows that the threshold of $1.5n$ cannot be increased. As a corollary, we show that any transitive $X$ satisfying $\limsup c_n(X) - n = \infty$ and $\limsup c_n(X) - 1.5n < \infty$ must be minimal. We also prove some restrictions on the structure of transitive non-minimal $X$ satisfying $\liminf c_n(X) - 2n = -\infty$, which imply unique ergodicity (for a periodic measure) as a corollary, which extends a result of Boshe…

The Furstenberg-Sárközy Theorem and Asymptotic Total Ergodicity Phenomena in Modular Rings

The Furstenberg-Sárközy theorem asserts that the difference set $E-E$ of a subset $E \subset \mathbb{N}$ with positive upper density intersects the image set of any polynomial $P \in \mathbb{Z}[n]$ for which $P(0)=0$. Furstenberg's approach relies on a correspondence principle and a polynomial version of the Poincaré recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,\mathcal{B},μ,T)$ and set $A \in \mathcal{B}$ with $μ(A) > 0$, one has $c(A):= \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N μ(A \cap T^{-P(n)}A) > 0.$ The limit $c(A)$…

无限区间交换变换上的斜积系统

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。

关于 $1$-有界乘性权的齐次遍历双线性平均

On the homogeneous ergodic bilinear averages with $1$-bounded multiplicative weights

本文推广了Bourgain双重回归定理和遍历Bourgain-Sarnak定理，证明非周期$1$-有界乘性权下齐次遍历双线性平均几乎处处收敛于零，并给出关键证明要素。

两类逐点遍历定理的统一方法：双重递归与返回时间

A Unified Approach to Two Pointwise Ergodic Theorems: Double Recurrence and Return Times

本研究提出统一方法，结合停时与度量熵，将Bourgain双重递归定理与返回时间定理推广到Kronecker序列整数部分，证明两类双线性遍历平均的几乎处处收敛性且收敛集对所有有理数γ一致。

乘性作用的分解结果及其应用

Decomposition results for multiplicative actions and applications

受齐次二次方程划分正则性问题启发，证明乘性保测作用的多次递归与收敛结果，核心方法是将有界可测函数分解为模拟pretentious与非周期乘性函数性质的分量，并利用新半范数及逆定理刻画系统因子。