Recurrence for stationary group actions

Spatial Recurrence for Ergodic Fractal Measures

We discuss an invertible version of Furstenberg's `Ergodic CP Shift Systems'. We show that the explicit regularity of these dynamical systems with respect to magnification of measures, implies certain regularity with respect to translation of measures; We show that the translation action on measures is non-singular, and prove pointwise discrete and continuous ergodic theorems for the translation action.

Directional recurrence and directional rigidity for infinite measure preserving actions of nilpotent lattices

Let $Γ$ be a lattice in a simply connected nilpotent Lie group $G$. Given an infinite measure preserving action $T$ of $Γ$ and a "direction" in $G$ (i.e. an element $θ$ of the projective space $P(\goth g)$ of the Lie algebra $\goth g$ of $G$), some notions of recurrence and rigidity for $T$ along $θ$ are introduced. It is shown that the set of recurrent directions $\Cal R(T)$ and the set of rigid directions for $T$ are both $G_δ$. In the case where $G=\Bbb R^d$ and $Γ=\Bbb Z^d$, we prove that (a) for each $G_δ$-subset $Δ$ of $P(\goth g)$ and a countable subset $D\subsetΔ$, there is a rank-one …

Recurrence on Affine Grassmannians

We study the action of the affine group $G$ of $\mathbb{R}^d$ on the space $X_{k,\,d}$ of $k$-dimensional affine subspaces. Given a compactly-supported Zariski dense probability measure $μ$ on $G$, we show that $X_{k,\,d}$ supports a $μ$-stationary measure $ν$ if and only if the $(k\!+\!1)^{\rm th}$-Lyapunov exponent of $μ$ is strictly negative. In particular, when $μ$ is symmetric, $ν$ exists if and only if $2k\geq d$.

Estimation of recurrence for nilpotent group action

We estimate size of recurrence of an action of a nilpotent group by homeomorphisms of a compact space for polynomial mappings into a nilpotent group form the partial semigroup $(\mathcal{P}_{f}(\mathbb{N}),\uplus)$. To do this we have used algebraic structure of the Stone-Čech copactification partial semigroup and that of the given nilpotent group.

Quantitative recurrence for free semigroup actions

We consider finitely generated free semigroup actions on a compact metric space and obtain quantitative information on Poincaré recurrence, average first return time and hitting frequency for the random orbits induced by the semigroup action. Besides, we relate the recurrence to balls with the rates of expansion of the semigroup's generators and the topological entropy of the semigroup action. Finally, we establish a partial variational principle and prove an ergodic optimization for this kind of dynamical action.

An extension of Furstenberg's structure theorem for Noetherian modules and multiple recurrence theorems III

Using a recent Furstenberg structure theorem, we obtain Multiple Recurrence Theorems relative to any locally compact second countable Noetherian module $G$ over a syndetic ring $R$, which generalizes Furstenberg's multiple recurrence theorem. In addition we study the multiple Birkhoff center and the pointwise multiple recurrence of a topological $G$-action on a compact metric space.

Rigidity, weak mixing, and recurrence in abelian groups

The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about $\mathbb{Z}$-actions extend to this setting: 1. If $(a_n)$ is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system. 2. There exists a sequence $(r_n)$ such that every translate is both a rigidity sequence and a set of recurrence. The first of these results was shown for $\mathbb{Z}$-actions by Adams, Fayad and Thouvenot, and…

Return times, recurrence densities and entropy for actions of some discrete amenable groups

Results of Wyner and Ziv and of Ornstein and Weiss show that if one observes the first k outputs of a finite-valued ergodic process, then the waiting time until this block appears again is almost surely asymptotic to $2^{hk}$, where $h$ is the entropy of the process. We examine this phenomenon when the allowed return times are restricted to some subset of times, and generalize the results to processes parameterized by other discrete amenable groups. We also obtain a uniform density version of the waiting time results: For a process on $s$ symbols, within a given realization, the density of the…

半分解的回归性

Recurrence for semi-decompositions

针对分解理论无法统一涵盖半群作用与滤过的问题，本文提出半分解概念，证明回归性相关关系可推广至该框架，并指出群作用与半群作用在回归概念上的差异。

非极大环面作用的刚性、幂单定量回复与丢番图逼近

Rigidity of non-maximal torus actions, unipotent quantitative recurrence, and Diophantine approximations

本文通过沿幂单方向的定量回复方法，给出了高阶对角化作用在 $SL_2$ 形式算术商上正熵测度的分类及其 Adelic 版本，并应用于整数倍丢番图逼近问题。

Non-stationary non-uniform hyperbolicity: SRB measures for dissipative maps

We prove the existence of SRB measures for diffeomorphisms where a positive volume set of initial conditions satisfy an "effective hyperbolicity" condition that guarantees certain recurrence conditions on the iterates of Lebesgue measure. We give examples of systems that do not admit a dominated splitting but can be shown to have SRB measures using our methods.

Recurrence for measurable semigroup actions

We study qualitative properties of the set of recurrent points of finitely generated free semigroups of measurable maps. In the case of a single generator the classical Poincare recurrence theorem shows that these properties are closely related to the presence of an invariant measure. Curious, but otherwise it turns out to be possible that almost all points are recurrent, while there is an wandering set of positive (non-invariant) measure. For a general semigroup the assumption about the common invariant measure for all generators looks somewhat unnatural (despite being widely used). Instead w…

Extended orbit properties on surfaces

In this paper, we study "demi-caractéristique" and (Poisson) stability in the sense of Poincaré. Using the definitions á la Poincaré for $\R$-actions $v$ on compact connected surfaces, we show that "$R$-closed" $\Rightarrow$ "pointwise almost periodicity (p.a.p.)" $\Rightarrow$ "recurrence" $\Rightarrow$ non-wandering. Moreover, we show that the action $v$ is "recurrence" with $|\mathrm{Sing}(v)| < \infty$ iff $v$ is regular non-wandering. If there are no locally dense orbits, then $v$ is "p.a.p." iff $v$ is "recurrence" without "orbits" containing infinitely singularities. If $|\mathrm{Sing}(…

The Structure of Strongly Stationary Systems

Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson introduced the notion of strong stationarity, showing that certain recurrence properties hold for arbitrary measure preserving systems if they are valid for strongly stationary ones. We construct some new examples and prove a structure theorem for strongly stationary systems. The building blocks are Bernoulli systems and rotations on nilmanifolds.

Characteristic factors for commuting actions of amenable groups

We describe characteristic factors for certain averages arising from commuting actions of locally compact, second-countable, amenable groups. Under some ergodicity assumptions we use these factors to prove a form of multiple recurrence for three such actions.

Stabilizers of Stationary Actions of Lattices in Semisimple Groups

Every stationary action of a strongly irreducible lattice or commensurator of such a latiice in a general semisimple group, with at least one higher-rank connected factor, either has finite stabilizers almost surely or finite index stabilizers almost surely.

遍历 $\mathbb{Z}^d$ 作用的原始平均、方向扩张性和定量扭曲回复性

Primitive Averages, Directional Expansivity, and Quantitative Twisted Recurrence for Ergodic $\mathbb{Z}^d$-Actions

本文为保概率 $\mathbb{Z}^d$ 作用建立了原始向量平均的 $L^2$ 极限公式，并给出了方向扩张集具有高下密度的谱条件，进而证明了扭曲多重回复定理的定量版本。

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…

递归性与不变测度的存在性

Recurrence and the existence of invariant measures

本文证明在描述集合论中，递归条件不保证不变Borel概率测度存在，并给出强化的反例构造。

Intersection spaces and multiple transverse recurrence

We study multiple recurrence properties along separated cross sections for pmp actions of unimodular lcsc group on Polish spaces. We establish a multiple transverse recurrence theorem under the assumption that sufficiently large powers of the return time set are Delone sets. Typical examples of such situations arise from the theory of uniform approximate lattices.

On chain recurrence classes of endomorphisms of $\mathbb P^k$

We prove that the minimal chain recurrence classes of a holomorphic endomorphism of $\mathbb P^k$ have finitely many connected components. We also obtain results on arbitrary classes. These strong constraints on the topological dynamics in the phase space are all deduced from the associated action on a space of currents.

Single recurrence in abelian groups

We collect problems on recurrence for measure preserving and topological actions of a countable abelian group, considering combinatorial versions of these problems as well. We solve one of these problems by constructing, in $G_{2}:=\bigoplus_{n=1}^{\infty} \mathbb Z/2\mathbb Z$, a set $S$ such that every translate of $S$ is a set of topological recurrence, while $S$ is not a set of measurable recurrence. This construction answers negatively a variant of the following question asked by several authors: if $A\subset \mathbb Z$ has positive upper Banach density, must $A-A$ contain a Bohr neighbor…

关于局部紧群的平稳作用及其Radon-Nikodym共轭函数

On stationary actions of locally compact groups and their Radon-Nikodym cocycles

研究局部紧群的平稳作用及其Radon-Nikodym共轭函数的结构与正则性，证明平稳作用非I型、非II_∞型，并建立通用的紧Radon-Nikodym模型，同时构造实仿射群上的随机游走，其Poisson边界不满足Kaimanovich的SAT*性质。

关于线性模式下的乘法递归

On multiplicative recurrence along linear patterns

本文完全刻画了满足乘法递归集必要条件的整数四元组$(a,b,c,d)$，推广了相关结果，并证明满足条件的集合$S$是乘法半群有限生成作用的递归集。

Multiple recurrence and large intersections for abelian group actions

The purpose of this paper is to study the phenomenon of large intersections in the framework of multiple recurrence for measure-preserving actions of countable abelian groups. Among other things, we show: (1) If $G$ is a countable abelian group and $φ, ψ: G \to G$ are homomorphisms such that $φ(G)$, $ψ(G)$, and $(ψ- φ)(G)$ have finite index in $G$, then for every ergodic measure-preserving system $(X, \mathcal{B}, μ, (T_g)_{g \in G})$, every set $A \in \mathcal{B}$, and every $\varepsilon > 0$, the set $\{g \in G : μ(A \cap T_{φ(g)}^{-1}A \cap T_{ψ(g)}^{-1}A) > μ(A)^3 - \varepsilon\}$ is synde…

圆周微分同胚群作用的维数理论 I：平稳测度

Dimension theory of group actions by circle diffeomorphisms I: Stationary measures

该论文研究圆周光滑随机游走平稳测度的维数性质，为同主题第二部分提供技术工具，还得到相关逼近定理与圆周作用时间反转对称性结论。

返回时间集与积递归

Return time sets and product recurrence

该研究探讨了在紧Hausdorff空间中，群作用下子集包含返回时间集的条件，并通过积递归与distality的关系回答了相关问题。

Stationary measures for $\mathrm{SL}_2(\mathbb{R})$-actions on homogeneous bundles over flag varieties

Let $G$ be a real semisimple Lie group with finite centre and without compact factors, $Q<G$ a parabolic subgroup and $X$ a homogeneous space of $G$ admitting an equivariant projection on the flag variety $G/Q$ with fibres given by copies of lattice quotients of a semisimple factor of $Q$. Given a probability measure $μ$, Zariski-dense in a copy of $H=\mathrm{SL}_2(\mathbb{R})$ in $G$, we give a description of $μ$-stationary probability measures on $X$ and prove corresponding equidistribution results. Contrary to the results of Benoist-Quint corresponding to the case $G=Q$, the type of station…

沿广义勾股数的矫饰系统的递归性

Recurrence for pretentious systems along generalized Pythagorean triples

针对沿广义勾股数（方程$ax^2 + b y^2 = c z^2$的解）的矫饰保测乘法作用建立多重递归结果，证实广义勾股分划正则性猜想的遍历理论形式，推论得相关有限着色自然数集含单色广义勾股数。

Reccurence Sets for Partial Inverse, Semigroup Actions and Related Structures

Two types of recurrence sets are introduced for inverse semigroup partial actions in topological spaces. We explore their connections with similar notions for related types of imperfect symmetries (prefix inverse semigroup expansions, partial group and groupoid actions).

Nonsingular Group Actions and Stationary S alpha S Random Fields

This paper deals with measurable stationary symmetric stable random fields indexed by R^d and their relationship with the ergodic theory of nonsingular R^d-actions. Based on the phenomenal work of Rosinski(2000), we establish extensions of some structure results of stationary S alpha S processes to SαS fields. Depending on the ergodic theoretical nature of the underlying action, we observe different behaviors of the extremes of the field.

平稳作用的无核原理与平稳随机子群的端

The No-Core Principle for Stationary Actions and Ends of Stationary Random Subgroups

本文证明了可数群平稳作用的无核原理，并将其应用于平稳随机子群的几何，证明了其Schreier图几乎必然有0、1、2或无穷多个端，并构造了具有任意端数的回旋镖子群。

On recurrence in zero-dimensional locally compact flow with compactly generated phase group

Let $X$ be a zero-dimensional locally compact Hausdorff space not necessarily metric and $G$ a compactly generated topological group not necessarily abelian or countable. We define recurrence at a point for any continuous action of $G$ on $X$, and then, show that if $\overline{Gx}$ is compact for all $x\in X$, the conditions (i) this dynamics is pointwise recurrent, (ii) $X$ is a union of $G$-minimal sets, (iii) the $G$-orbit closure relation is closed in $X\times X$, and (iv) $X\ni x\mapsto \overline{Gx}\in 2^X$ is continuous, are pairwise equivalent. Consequently, if this dynamics is distal,…

余紧群作用的无穷平稳测度

Infinite stationary measures of co-compact group actions

本文证明了局部紧 Hausdorff 空间上的余紧群作用存在非零平稳 Radon 测度，其核心方法是通过建立 Tarski 定理的平稳类比来完成证明。

Effective discreteness radius of stabilisers for stationary actions

We prove an effective variant of the Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a non-trivial intersection with a small $r$-neighborhood of the identity is at most $βr^δ$ for some explicit constants $β, δ> 0$ depending only the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a nove…

Invariant and stationary measures for the SL(2,R) action on Moduli space

We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner's seminal work.

半单李群平稳作用的Furstenberg熵谱

Furstenberg entropy spectra of stationary actions of semisimple Lie groups

本文通过精细化Nevo-Zimmer射影因子定理并在平稳随机子群上构造Poisson丛，完整确定了SL(d,R)及其格的遍历平稳作用的Furstenberg熵谱。

Closest integer polynomial multiple recurrence along shifted primes

Following an approach presented by N. Frantzikinakis, B. Host and B. Kra, we show that the parameters in the multidimensional Szemerédi theorem for closest integer polynomials have non-empty intersection with the set of shifted primes $\mathbb{P}-1$ (or similarly of $\mathbb{P}+1$). Using the Furstenberg Correspondence Principle, we show this result by recasting it as a polynomial multiple recurrence result in measure ergodic theory. Furthermore, we obtain integer part polynomial convergence results by the same method, which is a transference principle that enables one to deduce results for $\…

Generic Stationary Measures and Actions

Let $G$ be a countably infinite group, and let $μ$ be a generating probability measure on $G$. We study the space of $μ$-stationary Borel probability measures on a topological $G$ space, and in particular on $Z^G$, where $Z$ is any perfect Polish space. We also study the space of $μ$-stationary, measurable $G$-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak* topology. When $μ$ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of $(G,μ)$. When $Z$ is compact, this implies that the …

内作用在$C^*$-代数上的平稳态

Stationary states on a $C^*$-algebra for an inner action

本文研究可数离散群在$C^*$-代数上内作用的平稳态，证明其状态空间为Choquet单纯形，给出迹态与纯非迹态的分解及判别，并应用于自由群和性质(T)群，最后给出Bauer单纯形的充分条件。

Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions

We establish characterization results for the ergodicity of stationary symmetric $α$-stable (S$α$S) and $α$-Frechet random fields. We show that the result of Samorodnitsky [Ann. Probab. 33 (2005) 1782-1803] remains valid in the multiparameter setting, that is, a stationary S$α$S ($0<α<2$) random field is ergodic (or, equivalently, weakly mixing) if and only if it is generated by a null group action. Similar results are also established for max-stable random fields. The key ingredient is the adaption of a characterization of positive/null recurrence of group actions by Takahashi [Kodai Math. Se…

The Furstenberg-Zimmer structure theorem for stationary random walks

We prove the following version of the Furstenberg-Zimmer structure theorem for stationary actions: Any stationary action of a locally compact second-countable group is a weakly mixing extension of a measure-preserving distal system.

Topological Dynamics of Groupoid Actions

Some basic notions and results in Topological Dynamics are extended to continuous groupoid actions in topological spaces. We focus mainly on recurrence properties. Besides results that are analogous to the classical case of group actions, but which have to be put in the right setting, there are also new phenomena. Mostly for groupoids whose source map is not open (and there are many), some properties which were equivalent for group actions become distinct in this general framework; we illustrate this with various counterexamples.

Semigroup actions on tori and stationary measures on projective spaces

Let $Γ$ be a sub-semigroup of $G=GL(d,\mathbb R),$ $d>1.$ We assume that the action of $Γ$ on $\R^d$ is strongly irreducible and that $Γ$ contains a proximal and expanding element. We describe contraction properties of the dynamics of $Γ$ on $\R^d$ at infinity. This amounts to the consideration of the action of $Γ$ on some compact homogeneous spaces of $G,$ which are extensions of the projective space $\pr^{d-1}.$ In the case where $Γ$ is a sub-semigroup of $GL(d,\R)\cap M(d,\Z)$ and $Γ$ has the above properties, we deduce that the $Γ$-orbits on $\T^d=\R^d\slash\Z^d$ are finite or dense.

Weak equivalence of stationary actions and the entropy realization problem

We introduce the notion of weak containment for stationary actions of a countable group and define a natural topology on the space of weak equivalence classes. We prove that Furstenberg entropy is an invariant of weak equivalence, and moreover that it descends to a continuous function on the space of weak equivalence classes.

关于PSU(1,1)作用下S¹上平稳测度的复分析方法

On a complex-analytic approach to stationary measures on $S^1$ with respect to the action of $PSU(1,1)$

该研究利用广义解析延拓视角的柯西变换，提出PSU(1,1)作用下S¹平稳测度的复分析分类方法，改进Bourgain结果并借助Brown-Shields-Zeller定理完整刻画第一类富克斯群的Furstenberg测度。

Product-form stationary distributions for deficiency zero chemical reaction networks

We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg's deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balan…

Additive averages of multiplicative correlation sequences and applications

We study sets of recurrence, in both measurable and topological settings, for actions of $(\mathbb{N},\times)$ and $(\mathbb{Q}^{>0},\times)$. In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form $\{(an+b)^{\ell}/(cn+d)^{\ell}: n \in \mathbb{N}\}$ are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.

Twisted Recurrence via Polynomial Walks

In this paper we show how polynomial walks can be used to establish a twisted recurrence for sets of positive density in $\mathbb{Z}^d$. In particular, we prove that if $Γ\leq \operatorname{GL}_d(\mathbb{Z})$ is finitely generated by unipotents and acts irreducibly on $\mathbb{R}^d$, then for any set $B \subset \mathbb{Z}^d$ of positive density, there exists $k \geq 1$ such that for any $v \in k \mathbb{Z}^d$ one can find $γ\in Γ$ with $γv \in B - B$. Our method does not require the linearity of the action, and we prove a twisted recurrence for semigroups of maps from $\mathbb{Z}^d$ to $\mathb…

概率群中的混合性与双重回复

Mixing and double recurrence in probability groups

该研究定义了包含紧群、局部紧幺模顺从群超积的概率群类，发展其保测作用与混合性理论，证明混合性蕴含双重回复并给出定量版本，推广强化了已有结果，还为Furstenberg对应原理提供替代方案。