On Problems Of ?-expansions And Continued Fractions

In this dissertation, we focus on a problem of multifractal analysis in ?-dynamical system, the moving recurrent for doubling map and the size of a set in continued fractions whose partial quotients obey some relative conditions. We calculated the sizes of the related fractal sets in the sense of measure and Hausdorff dimension. This dissertation is divided into six chapter. In the first two chapters, we introduce the background and elementary preliminaries of this dissertation. And the following three chapters will discuss the three issues in detail.In the third chapter, we study the structure of ?-divergence points in the ?-dynamical system ([0,1),T?), where the Birkhoff averages of ?, do not exist. More precisely, the Hausdorff dimensions of the following multifractal decomposition sets were determined. where A(?,x) denotes all the accumulation points ofIn the fourth chapter, let (X, T, B,?) be a measure-theoretical dynamical system with a compatible metric d, a point x?X is called{nk}-moving recurrent if where{nk}k?N is a given sequence of integers. In this chapter, we prove that the set of{nk}-moving recurrent points is of full measure under any 2-fold mixing systems. And quantify the size of the set of {nk}-moving recurrent points in the sense of measure and Hausdorff dimension. We restrict our attention to the doubling map for convenience. Moreover, the results can be generalized to ?-expansions for all ?>1 with no more additional results.In the fifth chapter, we compute the Hausdorff dimension of the localized uniformly Jarnik set, which is defined as where is the sequence of the convergents of x ?[0,1] and ?:[0,1]?(0,+?) is a continuous function.At last, in the sixth chapter, we summarize the main results of this dissertation, and list some questions for further study.