Quantitative recurrence properties for self-conformal sets

TL;DR

This paper investigates the measure-theoretic recurrence properties of self-conformal sets under a shift map, establishing zero or full measure results based on the divergence or convergence of a volume sum.

Contribution

It provides a precise criterion for the measure of recurrence sets in self-conformal sets, linking it to volume sum divergence or convergence.

Findings

Recurrence sets have zero measure if the volume sum converges.

Recurrence sets have full measure under the open set condition if the volume sum diverges.

The results connect recurrence properties with classical volume sum criteria.

Abstract

In this paper we study the quantitative recurrence properties of self-conformal sets $X$ equipped with the map $T:X\to X$ induced by the left shift. In particular, given a function $\varphi:\mathbb{N}\to(0,\infty),$ we study the metric properties of the set $$R(T,\varphi)=\left\{x\in X:|T^nx-x|<\varphi(n)\textrm{ for infinitely many }n\in \mathbb{N}\right\}.$$ Our main result shows that for the natural measure supported on $X$, $R(T,\varphi)$ has zero measure if a natural volume sum converges, and under the open set condition $R(T,\varphi)$ has full measure if this volume sum diverges.