Quantitative recurrence properties in conformal iterated function systems

Abstract

Let $\Lambda$ be a countable index set and $S=\{\phi_i: i\in \Lambda\}$ 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 $$R(f):=\{x\in J^\ast: |T^nx-x|<e^{-S_n f(x)}, \ {\text{for infinitely many}}\ n\in \mathbb{N}\},$$ 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 $\inf\{s\ge 0: P(T, -s(f+\log |T'|))\le 0\}$, where $P$ is the pressure function and $T'$ is the derivative of $T$. We present some applications of the main theorem to Diophantine approximation.