Modified shrinking target problems on self-conformal sets

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, phi) := x is an element of J : ITnx -f(x)I < ?(n) for infinitely many n is an element of N , where f : J-+ J is a Lipschitz function and ? : N-+ R+ is a positive function defined on N. The Hausdorff dimension and zero-one law on the mu-measure of R(f, ?) are completely obtained, where mu stands for the natural self-conformal measure supported on J. (c) 2022 Elsevier Inc. All rights reserved.