Quantitative recurrence properties for self-conformal sets

Simon Baker; Michael Farmer

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arXiv · 1909.08913

Quantitative recurrence properties for self-conformal sets

Simon Baker·Michael Farmer

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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.

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BibTeX RIS

Simon Baker, Michael Farmer. 2019-09-19. Quantitative recurrence properties for self-conformal sets. https://arxiv.org/abs/1909.08913

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