Benoit Saussol
For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return back close to its starting point. We prove that when the decay of correlation is super-polynomial the recurrence rates and the pointwise dimensions are equal. This gives a broad class of systems for which the recurrence rate equals the Hausdorff dimension of the invariant measure.
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We study the lower and upper local rates of Poincare recurrence of rotations on the circle by means of symbolic dynamics. As a consequence, we show that if the lower rate of Poincare recurrence of an ergodic dynamical system (X,F,mu,T) is greater or equal to 1 mu-almost everywhere, then it is weakly mixing.
We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using $\ep$-chains, and the minimal lengths of these $\ep$-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.
We show that every invertible strong mixing transformation on a Lebesgue space has strictly over-recurrent sets. Also, we give an explicit procedure for constructing strong mixing transformations with no under-recurrent sets. This answers both parts of a question of V. Bergelson. We define $ε$-over-recurrence and show that given $ε> 0$, any ergodic measure preserving invertible transformation (including discrete spectrum) has $ε$-over-recurrent sets of arbitrarily small measure. Discrete spectrum transformations and rotations do not have over-recurrent sets, but we construct a weak mixing rigi…
Strong Positive recurrence for potential and exponential mixing of equilibrium states of surface diffeomorphisms
本文研究了具有正熵的表面微分同胚的大类势能的强正回复性，并建立了平衡态的指数衰减相关函数和有效内在遍历性。
We consider invertible discrete-time dynamical systems having a hyperbolic product structure in some region of the phase space with infinitely many branches and variable recurrence time. We show that the decay of correlations of the SRB measure associated to that hyperbolic structure is related to the tail of the recurrence times. We also give sufficient conditions for the validity of the Central Limit Theorem. This generalizes previous results by Benedicks and Young.
Let $(X, T)$ be a weakly mixing minimal system, $p_1, \cdots, p_d$ be integer-valued generalized polynomials and $(p_1,p_2,\cdots,p_d)$ be non-degenerate. Then there exists a residual subset $X_0$ of $X$ such that for all $x\in X_0$ $$\{ (T^{p_1(n)}x, \cdots, T^{p_d(n)}x): n\in \mathbb{Z}\}$$ is dense in $X^d$.
Strong positive recurrence and exponential mixing for diffeomorphisms
本研究引入闭流形上微分同胚的强正递归（SPR）性质，证明其可通过具谱隙的可数状态马尔可夫移位编码，推导最大熵测度的多种遍历性质，验证正熵C∞曲面微分同胚及更多低正则、高维情形满足SPR。
We prove a local central limit theorem for "nonconventional" sums generated by some classes of sufficiently fast mixing sequences.
We show that a non-wandering dynamical system with the shadowing property is either equicontinuous or has positive entropy and that in this context uniformly positive entropy is equivalent to weak mixing. We also show that weak mixing together with the shadowing property imply the specification property with a special kind of regularity in tracing (a weaker version of periodic specification property). This in turn implies that the set of ergodic measures supported on the closures of orbits of regularly recurrent points is dense in the space of all invariant measures (in particular, invariant m…
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…
We study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space X, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets \{n_k\} such that n_{k+1}/{n_k} tends to infinity, or such that n_{k} divides n_{k+1} for each k, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, fo…
Misiurewicz[19]introducedtheconceptofpseudo-entropyandproved this quantity coincides with topological entropy. Richeson et al. [21] obtained the lower bounded of topological entropy by means of the definition of pseudo-entropy. This paper aims to generalize the main results obtained by Misiurewicz and Rich-eson et al. to free semigroup actions. Firstly, the pseudo-entropy is introduced for free semigroup actions, and it is shown that the pseudo-entropy coincides with the topological entropy defined by Bufetov [9]. Secondly, these concepts of the chain recurrence, the chain mixing, the chain re…
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.
We prove a maximal-type large deviation principle for dynamical systems with arbitrarily slow polynomial mixing rates. Also several applications, particularly to billiard systems, are presented.
A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochastic-like behaviour such as large deviations or decay of correlations. Such geometric structures are generally highly non-trivial and thus a natural question is the extent to which this approach can be applied. In this paper we show that in many cases stochastic-like behaviour itself implies that the system has certain non-trivial geometric properties, which are therefore necessary as well as sufficient conditions for the…
This paper is a first step in the study of the recurrence behavior in random dynamical systems and randomly perturbed dynamical systems. In particular we define a concept of quenched and annealed return times for systems generated by the composition of random maps. We moreover prove that for super-polynomially mixing systems, the random recurrence rate is equal to the local dimension of the stationary measure.
We study Poincaré recurrence for flows and observations of flows. For Anosov flow, we prove that the recurrence rates are linked to the local dimension of the invariant measure. More generally, we give for the recurrence rates for the observation an upper bound depending on the push-forward measure. When the flow is metrically isomorphic to a suspension flow for which the dynamic on the base is rapidly mixing, we prove the existence of a lower bound for the recurrence rates for the observation. We apply these results to the geodesic flow and we compute the recurrence rates for a particular obs…
Two properties of a dynamical system, rigidity and non-recurrence, are examined in detail. The ultimate aim is to characterize the sequences along which these properties do or do not occur for different classes of transformations. The main focus in this article is to characterize explicitly the structural properties of sequences which can be rigidity sequences or non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations generally and for weakly mixing transformations in particular there are both parallels and distinctions between the class of rigid sequences …
From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes with two new results: (1) for $ϕ$-mixing and $ψ$-mixing processes, we give a new exponential approximat…
A topological dynamical system induces two natural systems, one is on the hyperspace and the other one is on the probability space. The connection among some dynamical properties on the original space and on the induced spaces are investigated. Particularly, a minimal weakly mixing system which induces a $P$-system on the probability space is constructed and some disjointness result is obtained.
We construct a set $S$ such that every translate of $S$ is a set of recurrence and a set of rigidity for a weak mixing measure preserving system. This construction generalizes or strengthens results of Katznelson, Saeki, Forrest, and Fayad and Kanigowski. The construction provides a density analogue of Julia Wolf's results on popular differences in finite abelian groups.
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。
Lecture notes on ergodic transformations
本讲义系统介绍了遍历变换的核心理论，涵盖遍历性等价性质、Birkhoff定理、弱混合、典型熵不变量、Poisson悬浮及酉算子谱定理等关键主题。
We prove that if a system has superpolynomial (faster than any power law) decay of correlations then the time $τ_{r}(x,x_{0})$ needed for a typical point $x$ to enter for the first time a ball $B(x_{0},r)$ centered in $x_{0},$ with small radius $r$ scales as the local dimension at $x_{0},$ i.e. $$\underset{r\to 0}{\lim}\frac{\log τ_{r}(x,x_{0})}{-\log r}=d_{μ}(x_{0}).$$
Rates of mixing for the measure of maximal entropy of dispersing billiard maps
本文研究有限视界色散台球映射最大熵测度的统计性质，在稍强的稀疏回归条件下证明其相关性至少多项式衰减并满足中心极限定理，且对有界复杂度台球桌给出超多项式衰减速率。
We prove that if a system has superpolynomial (faster than any power law) decay of correlations (with respect to Lipschitz observables) then the time $τ(x,S_{r})$ needed for a typical point $x$ to enter for the first time a set $S_{r}=\{x:f(x)\leq r\}$ which is a sublevel of a Lipschitz funcion $f$ scales as $\frac{1}{μ(S_{r})}$ i.e. \begin{equation*} \underset{r\to 0}{\lim }\frac{\log τ(x,S_{r})}{-\log r}=\underset{r\to 0}{\lim}\frac{\log μ(S_{r})}{\log (r)}. \end{equation*} This generalizes a previous result obtained for balls. We will also consider relations with the return time distributio…
Zero-one laws for eventually always hitting points in in rapidly mixing systems
该研究聚焦收缩目标系统中的最终始终击中点集合，针对伯努利概型、高斯映射这类目标平移近乎完全独立的系统，给出目标收缩速率的严格条件，判定该集合为零集或余零集。
Shrinking Targets versus Recurrence: a brief survey
本文综述了保测动力系统中收缩目标集与递归集在混合条件下的满测度结果，探讨了满测度命题的定量强化及理论关键差异。
We explore recurrence properties arising from dynamical approach to combinatorial problems like the van der Waerden Theorem. We describe relations between these properties, study their consequences for dynamics, and demonstrate connections to combinatorial problems. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity in topological weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.
Exponential mixing of measures of maximal entropy for certain skew products
本文研究了某些斜积中最大熵测度的指数混合性，通过纤维熵与李雅普诺夫指数的连续性关系，证明了满足强正回复性质的斜积具有有限数量的最大熵测度，且这些测度具有良好的统计性质。
The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system of finite entropy the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere. In addition the measure of $n$-cylinders is in various settings known to be lognormally distributed in the limit. In this paper the logarithm of the measure of $n$-cylinder, the information function, satisfies the almost sure invariance principle in the case in which the measure is $β$-mixing. We get a similar result for the recurrence time. Previous results are due to Ph…
We establish a new mixing theorem for quasirandom groups (finite groups with no low-dimensional unitary representations) $G$ which, informally speaking, asserts that if $g, x$ are drawn uniformly at random from $G$, then the quadruple $(g,x,gx,xg)$ behaves like a random tuple in $G^4$, subject to the obvious constraint that $gx$ and $xg$ are conjugate to each other. The proof is non-elementary, proceeding by first using an ultraproduct construction to replace the finitary claim on quasirandom groups with an infinitary analogue concerning a limiting group object that we call an \emph{ultra quas…
For any dynamical system $T:X\rightarrow X$ of a compact metric space $X$ with $g-$almost product property and uniform separation property, under the assumptions that the periodic points are dense in $X$ and the periodic measures are dense in the space of invariant measures, we distinguish various periodic-like recurrences and find that they all carry full topological topological entropy and so do their gap-sets. In particular, this implies that any two kind of periodic-like recurrences are essentially different. Moreover, we coordinate periodic-like recurrences with (ir)regularity and obtain …
We consider random perturbations of non-uniformly expanding maps, possibly having a non-degenerate critical set. We prove that, if the Lebesgue measure of the set of points failing the non-uniform expansion or the slow recurrence to the critical set at a certain time, for almost all random orbits, decays in a (stretched) exponential fashion, then the decay of correlations along random orbits is stretched exponential, up to some waiting time. As applications, we obtain almost sure stretched exponential decay of random correlations for Viana maps, as for a class of non-uniformly expanding local …
A pointwise ergodic theorem along return times of rapidly mixing systems
该研究引入快速相关衰减系统中由返回/到达时间随机生成的稀疏序列，证明其遍历且对遍历平均逐点普遍$L^2$-良好，为Bourgain返回时间定理的自然变体，还通过例子验证并适配非独立情形技术。
A strong Borel--Cantelli lemma for recurrence
本文研究混合动力系统中的回复性问题，通过构造满足测度条件的球序列，建立了强Borel–Cantelli引理，证明回复次数近似等于给定序列的部分和，并由此推出返回时间关于测度的对数增长率几乎处处为1。
We show that in a rapidly mixing flow with an invariant measure, the time which is needed to hit a given section is related to a sort of conditional dimension of the measure at the section. The result is applied to the geodesic flow of compact variable negative sectional curvature manifolds, establishing a logarithm law for such kind of flow.
Hitting statistics for $ϕ$-mixing dynamical systems
本文研究 $\phi$-混合动力系统中击中率与逃逸率的一般极限，证明其在无短返回时为一，否则由极值指数公式刻画，并适用于双曲、扩张及具 Young tower 的系统。
The shrinking target and recurrence problem for non-autonomous systems
本文研究了非自治系统中的收缩目标和复发问题，建立了零一准则，并通过统一框架分析了自治与非自治动态系统的行为。
Higher Order Wiener-Wintner systems: examples and applications
本研究构建高维Wiener-Wintner动力系统，通过扩展Bourgain的双重回归估计实现多重遍历平均的定量控制，给出多重回归相关平均几乎处处收敛的简洁证明，并提供Host-Kra-Ziegler因子的新特征。
Separating measurable recurrence from strong recurrence via rigidity sequences
本文通过构造测度扩张集的子集作为刚性序列，证明了对于几乎所有可数阿贝尔群，可测递归与强递归可被严格区分，并证实了 Ackelsberg 猜想。
Stochastic functional partial differential equations with monotone coefficients: Poisson stability measures, exponential mixing and limit theorems
针对单调系数随机泛函偏微分方程，本文利用耗散性与Shcherbakov比较法证明入口测度的存在唯一性及回复性继承，并建立极限点的一致指数混合、极限定理与收敛速率。
Chaotic multiscale dynamical systems are common in many areas of science, one of the examples being the interaction of the low-frequency dynamics in the atmosphere with the fast turbulent weather dynamics. One of the key questions about chaotic multiscale systems is how the fast dynamics affects chaos at the slow variables, and, therefore, impacts uncertainty and predictability of the slow dynamics. Here we demonstrate that the linear slow-fast coupling with the total energy conservation property promotes the suppression of chaos at the slow variables through the rapid mixing at the fast varia…
Let $Σ_{A}$ be a topologically mixing shift of finite type, let $σ:Σ_{A}\toΣ_{A}$ be the usual left-shift, and let $μ$ be the Gibbs measure for a Hölder continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system $(Σ_{A},σ)$ that hold $μ$-almost surely. In particular, given a function $ψ:\mathbb{N}\to \mathbb{N}$ we are interested in the following set $$R_ψ=\{{\texttt i}\in Σ_{A}:i_{n+1}\ldots i_{n+ψ(n)+1}=i_1\ldots i_{ψ(n)}\textrm{ for infinitely many }n\in\mathbb{N}\}.$$ We provide sufficient conditions for $μ(R_ψ)=1$ and suff…
This article reviews a generous sampling of both classical and more recent results on the interplay between measurable and topological dynamics. In the first part we have surveyed the strong analogies between ergodic theory and topological dynamics as shown in the treatment of recurrence phenomena, equicontinuity and weak mixing, distality and entropy. The prototypical result of the second part is the statement that any abstract measure probability preserving system can be represented as a continuous transformation of a compact space, and thus in some sense ergodic theory embeds into topologic…
Quantitative recurrence properties for piecewise expanding maps on $ [0,1]^d $
本文研究 $[0,1]^d$ 上分段扩张映射的定量回复性质，在指数混合和密度正则条件下证明了回复集测度由目标集体积之和的收敛性决定，并推广了先前矩阵变换的结果。
We establish exponential mixing for the geodesic flow $φ_t\colon T^1S\to T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal M}_{0,4}$ are exponentially mixing, in sharp contrast to the flows for ${\mathcal M}_{g,n}$ with $3g-3+n>1$, which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the pha…
Long-time behaviour of dynamical systems driven by bounded mixing noises
研究由有界混合噪声驱动的动力系统，通过将连续时间系统简化为离散时间随机动力系统，引入一类满足特定假设的混合随机力，在一定假设下证明有限维和无限维相空间的指数混合，还给出了对常微分方程和大气动力学偏微分方程的应用。
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