Quantitative recurrence properties for self-conformal sets

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 $\ddot{I}†:\mathbb{N}\to (0,\infty ),$ we study the metric properties of the set $$R(T,\ddot{I}†)=\left\{x\in X:|T^{n}x-x|<\ddot{I}†(n)\text{ for infinitely many }n\in \mathbb{N}\right\}.$$ Our main result shows that for the natural measure supported on $X$, $R(T,\ddot{I}†)$ has zero measure if a natural volume sum converges, and under the open set condition $R(T,\ddot{I}†)$ has full measure if this volume sum diverges.