Quantitative recurrence properties for self-conformal sets

Baker, Simon; Farmer, Michael

doi:10.1090/proc/15285

Quantitative recurrence properties for self-conformal sets
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by Simon Baker and Michael Farmer PDF

Proc. Amer. Math. Soc. 149 (2021), 1127-1138 Request permission

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 \begin{equation*} R(T,\varphi )=\left \{x\in X:|T^nx-x|<\varphi (n)\text { for infinitely many }n\in \mathbb {N}\right \}. \end{equation*} 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.

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