Quantitative recurrence properties for self-conformal sets,Proceedings of the American Mathematical Society

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Quantitative recurrence properties for self-conformal sets
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2021-01-21 , DOI: 10.1090/proc/15285
Simon Baker , Michael Farmer

Abstract:In this paper we study the quantitative recurrence properties of self-conformal sets

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

$\displaystyle R(T,\varphi )=\left \{x\in X:\vert T^nx-x\vert<\varphi (n)\text { for infinitely many }n\in \mathbb{N}\right \}.$

Our main result shows that for the natural measure supported on

, $R(T,\varphi )$ 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.

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