Quantitative recurrence properties for self-conformal sets

Abstract

In this paper we study the quantitative recurrence properties of self-conformal sets X X equipped with the map T : X → X T:X\to X induced by the left shift. In particular, given a function φ : N → ( 0 , ∞ ) , \varphi :\mathbb {N}\to (0,\infty ), we study the metric properties of the set R ( T , φ ) = { x ∈ X : | T n x − x | > φ ( n ) for infinitely many n ∈ N } . \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 X , R ( T , φ ) R(T,\varphi ) has zero measure if a natural volume sum converges, and under the open set condition R ( T , φ ) R(T,\varphi ) has full measure if this volume sum diverges.