Quantitative recurrence properties for piecewise expanding maps on $ [0,1]^d $

TL;DR

This paper studies recurrence properties of piecewise expanding maps on multi-dimensional spaces, establishing measure-zero or full results based on the sum of target set volumes, extending previous work to more general transformations.

Contribution

It extends recurrence results to non-integer matrices and more general target sets, and provides a dimension result for diagonal matrix transformations.

Findings

Measure of recurrence set is zero or full depending on volume sum.

Results apply to non-integer matrix transformations.

Dimension results obtained for diagonal matrices.

Abstract

Let $ T\colon[0,1]^d\to [0,1]^d $ be a piecewise expanding map with an absolutely continuous invariant measure $ \mu $. Let $ \{H_n\} $ be a sequence of hyperrectangles or hyperboloids centered at the origin. Denote by $ \mathcal R(\{H_n\}) $ the set of points $ \mathbf x $ such that $ T^n\mathbf x\in \mathbf x+H_n $ for infinitely many $ n\in\mathbb N $, where $ \mathbf x+H_n $ is the translation of $ H_n $. We prove that if $ \mu $ is exponential mixing and the density of $ \mu $ is sufficiently regular, then the $\mu$-measure of $ \mathcal R(\{H_n\}) $ is zero or full according to the sum of the volumes of $ H_n $ converges or not. In the case that $ T $ is a matrix transformation, our results extend a previous work of Kirsebom, Kunde, and Persson [to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., 2023] in two aspects: by allowing the matrix to be non-integer and by allowing the `target' sets $ H_n $ to be hyperrectangles or hyperboloids. We also obtain a dimension result when $ T $ is a diagonal matrix transformation.