Metric Theory Of Some Recurrent Sets Arising In Conformal Iterated Function Systems

This dissertation mainly discusses the metric theory of some recurrent sets arising in conformal iterated function systems.In this dissertation,we first investigate a modified version of the shrin_king target problem on self-conformal sets,which generated by conformal iterated function systems.Secondly,we investigate a moving recurrent problem for the nonautonomous dynamical systems induced by Cantor series expansions.The whole dissertation is divided into five chapters,among which first two chapters are devoted to providing the relevant backgrounds and basic knowledge required for the research issues.In the next two chapters,we will talk about the two main problems of this dissertation in detail.In the third chapter,we investigate a modified version of the shrin_king target problem on self-conformal sets,which unifies the classic shrin_king 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 many n ∈N},where f:J→J is a Lipschitz map and φ:N→R⁺ is a positive function.The Hausdorff dimension and zero-one law on the μ-measure of R（f,φ）are completely obtained,where μstands for the natural self-conformal measure supported on J.In the fourth chapter,we investigate a moving recurrent problem for the nonautonomous dynamical system induced by Cantor series expansion.Let Q={q_k}_k≥1 be a sequence of positive integers with q_k≥ 2 for all k≥1.Put T_Qⁿ（x）= q₁…q_nx-[q₁…q_nx]for each n≥1,which gives the Q-Cantor series expansion.We focus on the following{n_k,r_k}-moving recurrent points proposed by Boshernitzan&Glasner:（?）,where {n_k}_k≥1 and {r_k}_k≥1 are two given sequences of positive integers.It is proved that when {n_k}_k≥1 and {r_k}_k≥1 tend to infinity,the set of {n_k,r_k}-moving recurrent points is of full Lebesgue measure.In addition,we study the size of the following quantitative version of {n_k,r_k}-moving recurrent set:R（{n_k,r_k}）:={x ∈[0,1]:|T_Q^nk（x）-T_Q^n_k+r_k（x）|<φ（k）for infinitely many k∈N},where φ:N→R⁺ is a positive function.It is proved that under the additional condition that {n_k+r_k}_k≥1 is strictly increasing,the Lebesgue measure and Hausdorff measure of R（{n_k,r_k}）respectively fulfill a dichotomy law according to the convergence or divergence of certain series.In the last chapter,we summarize our main results appearing in this dissertation and propose some questions for further study.