The Study On Moving Recurrent Properties For The Doubling Map

Let (X, T,B, u) be a measure-theoretical dynamical system with a compatible metric d. Following Boshernitzan, call a point x ?X is{nk}-moving recurrent if where{nk}k?N is a given sequence of integers. It was asked whether the set of{nk}-moving recurrent points is of full ?-measure. In this thesis, we restrict our attention to the doubling map and quantify the size of the set of{nk}-moving recurrent points in the sense of measure and Hausdorff dimension.The first chapter is introduction, mainly introducing the research background and its significance, and outlining some researches and the relevant conclusions on this issue in domestic and foreign. In chapter 2, we introduce some relevant prior knowledge, mainly including the dyadic expansions, the definition of k-mixing system and two inequality lemma on the dimension of a set. In chapter 3, in order to prove the theorem 1.1, at first, we construct a subset of target set; secondly, we conclude the result that the measure of the set is 1 by using the method of apagoge and the definition of k-mixing system; thirdly, estimate the measure of the set of{nk}-moving recurrent points is 1 on the unit interval. In chapter 4, mainly prove the Hausdorff dimension of the following set By considering the natural coverings of R({nk,rk}), we obtain the upper bound; by constructing a subset of Cantor set in R({nk,rk}) and using the dimensional result, we obtain the lower bound. In chapter 5, mainly study the Hausdorff dimension of the set of non{nk}-moving recurrent points. We construct a subset of the target set. Then we can get the fact that the Hausdorff dimension of the subset is 1 by applying dimensional result. In the final chapter, we mainly discussed the related conclusion and promotion.