Quantitative recurrence properties and homogeneous self-similar sets

TL;DR

This paper investigates the recurrence behavior of points in homogeneous self-similar sets under a natural shift map, establishing a zero-one law for the measure of points recurrent within a shrinking neighborhood, based on series convergence.

Contribution

It provides a complete metric theory for recurrence sets in self-similar sets, including a dichotomy law for measure and Hausdorff measure, extending previous results in fractal dynamics.

Findings

Measure of recurrence set is zero or full depending on series convergence.

A similar dichotomy law applies to Hausdorff measure.

Completes the metric theory of recurrence in self-similar sets.

Abstract

Let $K$ be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map $T: K\rightarrow K$ induced by the shift. Let $\mu$ be the natural self-similar measure supported on $K$. For a positive function $\varphi$ defined on $\mathbb{N}$, we show that the $\mu$-measure of the following set \begin{equation*} R(\varphi):=\{x\in K: |T^n x-x|<\varphi(n) \; \text{for infinitely many} \; n\in\mathbb{N}\} \end{equation*} is null or full according to convergence or divergence of a certain series. Moreover, a similar dichotomy law holds for the general Hausdorff measure, which completes the metric theory of this set.