Quantitative recurrence properties in conformal iterated function systems

Abstract

Let $\Lambda$ be a countable index set and $S=\{{\varphi }_i:i\in \Lambda \}$ be a conformal iterated function system on $[0,1{]}^d$ satisfying the open set condition. Denote by $J$ the attractor of $S$. With each sequence $(w_1,w_2,...)\in {\Lambda }^{\mathbb{N}}$ is associated a unique point $x\in [0,1{]}^d$. Let $J^{\ast }$ denote the set of points of $J$ with unique coding, and define the mapping $T:J^{\ast }\to J^{\ast }$ by $Tx=T(w_1,w_2,w_3...)=(w_2,w_3,...)$. In this paper, we consider the quantitative recurrence properties related to the dynamical system $(J^{\ast },T)$. More precisely, let $f:[0,1{]}^d\to {\mathbb{R}}^{+}$ be a positive function and $$R(f):=\{x\in J^{\ast }:|T^{n}x-x|<e^{-S_{n}f(x)},\text{for infinitely many}n\in \mathbb{N}\},$$ where $S_{n}f(x)$ is the $n$th Birkhoff sum associated with the potential $f$. In other words, $R(f)$ contains the points $x$ whose orbits return close to $x$ infinitely often, with a rate varying along time. Under some conditions, we prove that the Hausdorff dimension of $R(f)$ is given by $\inf \{s\ge 0:P(T,-s(f+\log |T^{\prime }|))\le 0\}$, where $P$ is the pressure function and $T^{\prime }$ is the derivative of $T$. We present some applications of the main theorem to Diophantine approximation.

25 pages

References in corpus (2)

• Quantitative recurrence properties of expanding maps
• Shrinking Targets for Countable Markov Maps