In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero-one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero-one laws for the two problems.
A classical Borel Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will almost surely happen. In this article, we propose an extension of Borel Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions, Diophantine approximations and extreme value theory for dynamica…
We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system $(X, μ, T)$ with a compatible metric $d$. We prove that, under some regularity conditions, the $μ$-measure of the following set \[ R(ψ)= \{x\in X : d(T^n x, x) < ψ(n)\ \text{for infinitely many}\ n\in\N \} \] obeys a zero-full law according to the convergence or divergence of a certain series, where $ψ:\N\to\R^+$. Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.
Let $X\cong \operatorname{SL}_2(\mathbb R)/\operatorname{SL}_2(\mathbb Z)$ be the space of unimodular lattices in $\mathbb R^2$, and for any $r\ge 0$ denote by $K_r\subset X$ the set of lattices such that all its nonzero vectors have supremum norm at least $e^{-r}$. These are compact nested subset{s} of $X$, with $K_0 = {\bigcap}_{r}K_r$ being the union of two closed horocycles. We use an explicit second moment formula for the Siegel transform of the indicator functions of squares in $\mathbb R^2$ centered at the origin to derive an asymptotic formula for the volume of sets $K_r$ as $r\to 0$. …
We derive new variants of the quantitative Borel--Cantelli lemma and apply them to analysis of statistical properties for some dynamical systems. We consider intermittent maps of $(0,1]$ which have absolutely continuous invariant probability measures. In particular, we prove that every sequence of intervals with left endpoints uniformly separated from zero is the strong Borel--Cantelli sequence with respect to such map and invariant measure.
With a view to establishing measure theoretic approximation properties of Delone sets, we study a setup which arises naturally in the problem of averaging almost periodic functions along exponential sequences. In this setting, we establish a full converse of the Borel-Cantelli lemma. This provides an analogue of more classical problems in the metric theory of Diophantine approximation, but with the distance to the nearest integer function replaced by distance to an arbitrary Delone set.
This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincaré recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance prop…
