Diophantine Analysis Of The Orbits In The Dynamical Systems With Specification Property

This paper studies the problem of Diophantine analysis of the orbits in the dynamical systems with specification property. A general prospect is to find a unified principle for the fractal dimensions of related sets. In this paper,we prove to be the following conclusion in the following system:Let (X,d) be a compact metric space and T:X?X a continuous transformation. Assume that(X,T) has the specification property, then for any a? 0. Where htop denotes the topological entropy.The first chapter is introduction, and it mainly introduces the research background, and outlines the research status and the relevant conclusions on this issue in domestic and foreign, and involves the content of the study in this paper. In chapter 2, we introduce some relevant preliminary knowledge, mainly including the definition of specification property, topological entropy for compact sets and general sets, and entropy distribution principle and its proof. In the next chapter, we mainly prove the upper bound of theorem 1.3.. In chapter 4, in order to prove the lower bound of theorem 1.3., at first, for any ?> 0,we construct a Cantor-like set Ea,?; secondly, construct a probability measure ?? supported on Ea,?; and at last, we conclude the corresponding results by applying the entropy distribution principle. In the final chapter, we mainly pose our question or speculation about the dimension in a general system and the related conclusion and promotion.