Quantitative recurrence properties and homogeneous self-similar sets

Abstract

Let K be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map T : K → K induced by the shift. Let µ be the natural self-similar measure supported on K. For a positive function ϕ defined on N, we show that the µ-measure of the following set R(ϕ) := {x ∈ K : |T n x − x| < ϕ(n) for infinitely many n ∈ N} is null or full according to convergence or divergence of a certain series. Moreover, a similar dichotomy law holds for the general Hausdorff measure, which completes the metric theory of this set.2010 Mathematics Subject Classification. Primary 28A80; 28D05; Secondary 11K55.

On the Hausdorff Measure of Shrinking Target Sets on Self‐conformal Sets

“…In recent years, several other authors have also studied the problem of shrinking targets on fractals. For example, Chernov and Kleinbock [9] studied the measure of shrinking target sets with respect to ergodic measures, Chang, Wu, and Wu [8] very recently studied the problem of recurrence sets on linear IFSs consisting of maps with equal contraction ratios, Koivusalo and Ramírez [25] considered shrinking targets on self-affine sets, the second author and Rams computed the Hausdorff dimension for certain shrinking targets on Bedford-McMullen carpets [4], and Seuret and Wang considered some related problems in the setting of conformal IFSs [32]. However, the value of the dim H (W (x, ))-dimensional Hausdorff measure remained unknown except in some very special cases.…”

On the Hausdorff measure of shrinking target sets on self-conformal sets

“…In recent years, several other authors have also studied the problem of shrinking targets on fractals. For example, Chernov and Kleinbock [12] studied the measure of shrinking target sets with respect to ergodic measures, Chang, Wu and Wu [11] very recently studied the problem of shrinking targets on linear iterated iterated function systems consisting of maps with equal contraction ratios, Koivusalo and Ramírez [30] considered shrinking targets on self-affine sets, the second author and Rams computed the Hausdorff dimension for certain shrinking targets on Bedford-McMullen carpets [4], and Seuret and Wang considered some related problems in the setting of conformal iterated function systems [39].…”

Dynamical Borel-Cantelli lemma for recurrence theory

“…As far as a general error function ψ is concerned, hardly anything is known. The only known results for µ-measure of R(ψ) are recently proven by Chang-Wu-Wu [4], Baker-Farmer [1], and Kirsebom-Kunde-Persson [6]. Chang-Wu-Wu [4] considered homogeneous self similar set satisfying the strong separation condition.…”

“…The only known results for µ-measure of R(ψ) are recently proven by Chang-Wu-Wu [4], Baker-Farmer [1], and Kirsebom-Kunde-Persson [6]. Chang-Wu-Wu [4] considered homogeneous self similar set satisfying the strong separation condition. Baker-Farmer [1] generalised Chang-Wu-Wu's result to the finite conformal iterated function systems with open set condition.…”

“…Remark 1.6 (Chang-Wu-Wu [4], Baker-Farmer [1]). The results of Chang-Wu-Wu [4] and Baker-Farmer [1] are applicable to finite conformal iterated function systems with open set condition and the map T : X → X induced by the left shift. Generally speaking the set…”