Quantitative recurrence properties in conformal iterated function systems

Let $\mathrm{\Lambda }$ be a countable index set and $S=\{{\varphi }_i:i\in \mathrm{\Lambda }\}$ 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 $(w_1,w_2,\mathrm{.}\mathrm{.}\mathrm{.})\in {\mathrm{\Lambda }}^{\mathbb{N}}$ is associated a unique point $x\in [0,1]^d$\mathrm{.}Let$J^\ast$denotethesetofpointsof$J$ 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 $$R(f):=\{x\in J^{\ast }:\mathrm{\mid }{T}^{n}x-x\mathrm{\mid }<e^{-S_{n}f(x)},\text{ for infinitely many }n\in \mathbb{N}\},$$ 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$P$isthepressurefunctionand$T’$ is the derivative of $T$. We present some applications of the main theorem to Diophantine approximation.