S. Galatolo
Quantitative recurrence indicators are defined by measuring the first entrance time of the orbit of a point $x$ in a decreasing sequence of neighborhoods of another point $y$. It is proved that these recurrence indicators are a.e. greater or equal to the local dimension at $y$, then these recurrence indicators can be used to have a numerical upper bound on the local dimension of an invariant measure.
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Recurrence times, waiting times and universal entropy production estimators
本研究将基于递归与等待时间渐近性的通用典型信号估计方法推广至非平衡统计力学，提出并分析了有限字母表单侧移位系统中的熵产生估计量。
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…
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. $$\underset{r\to 0}{\lim}\frac{\log τ_{r}(x,x_{0})}{-\log r}=d_{μ}(x_{0}).$$
Let $β>1$ be a real number. In this paper, the Hausdorff dimension of sets consisting of pairs of numbers with prescribed quantitative waiting time indicators in $β$-expansions are determined. More precisely, let $I$ be the unit interval $[0,1)$ and write $\underline{R}^β(x,y)$ and $\overline{R}^β(x,y)$ as the lower and upper quantitative waiting time indicators of $y$ by $x$ in $β$-expansions, respectively. Define the waiting set on the plane by \[E_β(a,b)=\left\{(x,y)\in I^2\colon\underline{R}^β(x,y)=a,\overline{R}^β(x,y)=b\right\}.\] where $0\leq a\leq b\leq\infty$, then the set $E_β(a,b)$ …
In this article, we consider hyperbolic rational maps restricted on thier Julia sets and study about the recurrence rate of typical orbits in arbitrarily small neighbourhoods around them and their relationship to the Hausdorff dimension of such small neighbourhoods.
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.
Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighborhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, including $\ZZ^2$-extension of hyperbolic dynamics. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a convergence in distribution of the rescaled return times near the origin.
A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities, called an observation. Following this idea and in the continuity of Boshernitzan's work, for a measure preserving system, we study Poincaré recurrence for the observation. The link between the return time for the observation and the Hausdorff dimension of the image of the invariant measure is considered. We prove that when the decay of correlations is super polynomial, the recurrence rates for the observations and the pointwise dimensions relative…
We consider linear mappings on the $d$-dimensional torus, defined by $T(x) = Ax \pmod 1$, where $A$ is an invertible $d \times d$ integer matrix, with no eigenvalues on the unit circle. In the case $d = 2$ and $\det A = \pm 1$, we give a formula for the Hausdorff dimension of the set \[ \{ \, x \in \mathbb{T}^d : d (T^n (x), x) < e^{- αn} \text{ for infinitely many } n \, \}. \]
Quantitative recurrence problem on some Bedford-McMullen carpets
针对 Hausdorff 维数与盒维数相等的 Bedford-McMullen 地毯，论文研究其典范自同态定量回归集的 Hausdorff 维数。
We are interested in the asymptotic behaviour of the first return time of the orbits of a dynamical system into a small neighbourhood of their starting points. We study this quantity in the context of dynamical systems preserving an infinite measure. More precisely, we consider the case of $\mathbb{Z}$-extensions of subshifts of finite type. We also consider a toy probabilistic model to enlight the strategy of our proofs.
Shrinking targets versus recurrence: A survey on the Hausdorff dimension theory
该综述对比动力系统中收缩目标集与回归集的豪斯多夫维数，涉及多类动力系统，还涵盖相关测度结果及坏逼近轨道等内容。
Quantitative recurrence for Z-extension of three-dimensional Axiom A flows
本文研究黎曼流形上Axiom A流的Z-扩张的定量回复性，分析首次返回起点小邻域时间的渐近行为，建立几乎处处与依分布收敛结果，可应用于负曲率紧曲面Z-覆盖的测地流。
Scale Recurrence Lemma and Dimension Formula for Cantor Sets in the Complex Plane
本文将 Moreira-Yoccoz 尺度递归引理推广到复平面康托集的多维共形情形，并据此在适当假设下证明了乘积康托集在函数作用下 Hausdorff 维数的计算公式。
We discuss selected topics of current research interest in the theory of dynamical systems, with emphasis on dimension theory, multifractal analysis, and quantitative recurrence. The topics include the quantitative versus the qualitative behavior of Poincaré recurrence, the product structure of invariant measures and return times, the dimension of invariant sets and invariant measures, the complexity of the level sets of local quantities from the point of view of Hausdorff dimension, and the conditional variational principles as well as their applications to problems in number theory. We prese…
Estimating the Hausdorff measure by recurrence
本文提出一种通过计算点自身最近回归时间来从下方估计分形集豪斯多夫测度的新方法。
We study Poincaré recurrence from a purely geometrical viewpoint. We prove that the metric entropy is given by the exponential growth rate of return times to dynamical balls. This is the geometrical counterpart of Ornstein-Weiss theorem. Moreover, we show that minimal return times to dynamical balls grow linearly with respect to its length. Finally, some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures are given.
We investigate quantitative recurrence in systems having an infinite measure. We extend the Ornstein-Weiss theorem for a general class of infinite systems estimating return time in decreasing sequences of cylinders. Then we restrict to a class of one dimensional maps with indifferent fixed points and calculate quantitative recurrence in sequences of balls, obtaining that this is related to the behavior of the map near the fixed points.
We consider the billiard dynamics in a non-compact set of R^d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called `quenched random Lorentz tube'. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties.
Intersections of quantitative recurrence and badly approximable sets for expanding Markov maps
针对单位区间上含 sofic β-变换等的扩张马尔可夫映射，研究动态 badly approximable 点的定量回归性质，确定回归集与 badly approximable 集等交集的豪斯多夫维数，得到相关结果。
In this paper we prove that a class of skew products maps with non uniformly hyperbolic base has exponential decay of correlations. We apply this to obtain a logarithm law for the hitting time associated to a contracting Lorenz attractor at all the points having a well defined local dimension, and a quantitative recurrence estimation.
A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class
本文提出一种 $C^1$-扰动技术，用于在保留同宿/异宿交集的前提下，将给定周期点集从周期点的链回归类中弹出，并应用于三维部分双曲微分同胚设定。
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。
Hausdorff dimension of recurrence sets for matrix transformations of tori
本文研究环面上由整数矩阵诱导的变换的回归集的豪斯多夫维数，在矩阵可对角化且特征值为整数时给出了维数公式。
A counterexample to Lagrangian Poincaré recurrence in dimension four
本文通过近托里纤维化在四维空间中构造了Lagrangian庞加莱复发的反例，挑战了传统理论。
Large deviations for occupation and waiting times of infinite ergodic transformations
本文研究了保持无限测度的遍历变换的occupation和waiting时间的大偏差，采用双Laplace变换方法建立了与Darling-Kac定理和广义正弦定律相关的新估计。
More counterexamples to Lagrangian Poincaré recurrence in dimension four
本文扩展了拉格朗日庞加莱回归的反例族，使其覆盖所有非单调环面辛四流形，补充了作者早前关于环面辛四流形反例的研究。
We show that the Poincaré return time of a typical cylinder is at least its length. For one dimensional maps we express the Lyapunov exponent and dimension via return times.
We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases this set is equivalent to the recurring set on the fractal.
For a given rotation number we compute the Hausdorff dimension of the set of well approximable numbers. We use this result and an inhomogeneous version of Jarnik's theorem to show strong recurrence properties of the billiard flow in certain polygons
Various issues with regard to chaos and recurrence in infinite dimensions are discussed. The doctrine we are trying to derive is that Sobolev spaces over bounded spatial domains do host chaos and recurrence, while Sobolev spaces over unbounded spatial domains are lack of chaos and recurrence. Local Sobolev spaces over unbounded spatial domains can host chaos and are natural phase spaces e.g. for fluid problems, but are very challenging to study.
Many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks. In this work, we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional Brownian motions (FBMs). First, our results indicate that the constructed recurrence networks have exponential degree distributions; the relationship between $H$ and $<λ>$ can be represented by a cubic polynomial function. We next focus on the motif rank distribution of recurrence networks, so that we can better understand networks at…
Under a map T, a point x recurs at rate given by a sequence {r_n} near a point x_0 if d(T^n(x),x_0)< r_n infinitely often. Let us fix x_0, and consider the set of those x's. In this paper, we study the size of this set for expanding maps and obtain its measure and sharp lower bounds on its dimension involving the entropy of T, the local dimension near x_0 and the upper limit of 1/n log 1/r_n. We apply our results in several concrete examples including subshifts of finite type, Gauss transformation and inner functions.
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