Shrinking Target Problems For Beta Dynamical System

Let(X,T,?,d)be a measure theoretic dynamical system with a metric d.If T is ergodic with respect to the measure ?,Poincare recurrence theorem implies that,While,this is only qualitative in nature;it does not address either with which rate the orbit will return back to the initial point or in which manner the neighborhood of the initial point can shrink.In 1995,Hill and Velani[1]first introduced the shrinking targets theory.For x0 ? X,let?:N?R+,they determined the Hausdorff dimension of the following set S(T,?):={x?X:d(Tnx,X0)<?(n)for infinitely many n ? N}.Actually,there are three types of the shrinking target problems:shrinking target problems with given targets,covering problems and quantitative Poincare recurrence properties.In this paper,we focus on shrinking target problems for beta dynamical system.This paper is divided into six chapters.The first two chapters respectively introduce the relevant backgrounds,elementary preliminaries and our results of this paper.The next three chapters are the proof of our results.In the third chapter,we consider modified shrinking target problem in beta dynamical systems.We know that Tan,Wang[2]and Shen,Wang[3]calculated the Hausdorff dimension of the two sets R(T?,?)and S(T?,?)(see(1.1),(1.2))respectively,and they showed that the two sets have the same dimension formula.Therefore,we want to know whether there is a unified method to find the Hausdorff dimension of the two sets in the beta dynamic system.Therefore,this leads to a modified shrinking target problem.Define W(f,?)={x?[0,1):|TBnx-f(x)|<?(n)for infinitely many n ? N},where f:[0,1]?[0,1]is a Lipschitz function.We will determine the Hausdorff dimension of the set W(f,?).In the fourth chapter,we consider simultaneous dynamical Diophantine approximation in beta dynamical systems.Following the work of Hill and Velani[1],the Hausdorff dimension of the set S(T,?)has been determined for many dynamical systems.They have established a complete metrical theory.However,not much is known for the higher dimensional version.Let ?i(i=1,2)be two positive functions on N such that ?i(n)?0 when n??.Define W(T?,?1,?2)={(x,y)?[0,1]2:|T?nx-x0|<?1(n),|T?ny-y0|<?2(n)for infinitely many n ? N},where x0,y0?[0,1]are two given fixed points.In this chapter,We shall give the Lebesgue measure and the Hausdorff dimension of the set W(T?,?1,?2).In the fifth chapter,We change the error function of the set W(T?,?1,?2)·The shrinking rates depend upon the points to be approximated and hence naturally provide better approximation properties than the conventional positive error function ?.Dependence of the error functions on the points to be approximated significantly increases the level of difficulty.Define E(T?,f,g)={(x,y)?[0,1]2:|T?nx-x0|<e-Snf(x).|T?ny-y0|<e-Sng(y)for infinitely many n ? N},where Snf(x)=?j=0 n-1 f(T?jx).We prove the Hausdorff dimension of the set E(T?,f,g),which is related to the pressure function.At last,we summarize the main results of this paper.At the same time,we also put some shrinking target problems for further study.