Modified shrinking target problems on self-conformal sets

Abstract

In this paper, we investigate a modified version of the shrinking target problem on self-conformal sets, which unifies the shrinking target problems and quantitative recurrence properties. Let J be a self-conformal set generated by a conformal iterated function system satisfying the open set condition, and let T:J→J be the expanding map induced by the left shift. We will study the size of the following set:R(f,φ):={x∈J:|Tnx−f(x)|<φ(n)for infinitely manyn∈N}, where f:J→J is a Lipschitz function and φ:N→R+ is a positive function defined on N. The Hausdorff dimension and zero-one law on the μ-measure of R(f,φ) are completely obtained, where μ stands for the natural self-conformal measure supported on J.