Dimension and waiting time in rapidly mixing systems

关于Kontoyiannis的一个等待时间结果：混合还是解耦？

On a waiting-time result of Kontoyiannis: mixing or decoupling?

本文将下解耦条件引入两个独立平稳随机过程间交叉熵的等待时间估计，替代Kontoyiannis结果中的$ψ$-混合条件，大幅扩展了该估计的适用范围。

Recurrence rate in rapidly mixing dynamical systems

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.

Hitting time in regular sets and logarithm law for rapidly mixing dynamical systems

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…

快速混合系统中最终始终击中各点的零一律

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。

快速混合系统返回时间的逐点遍历定理

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。

Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds

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.