Distribution Of Full Cylinders In Beta Expansions

In1957, A. Renyi introduced the beta-expansion, for any real number β>1, which is a generalization of beta-expansion. On one hand, from the point of view of representa-tion theory of real numbers, beta-expansion greatly expands the collection of numbers with precise expansion. On the other hand, for a general beta, its corresponding dynamical sys-tem lacks of finite Markov property. This makes beta-expansion an important status in the representing of real numbers and dynamical system.In order to study the metric properties of beta-expansion, a basic problem is to give effective estimate on the length of a cylinder. As is known to all, for a general beta, if its corresponding dynamical system has finite Markov property, the distribution of cylin-ders with regular lengths and we can give effective estimate on the length of the cylinder. Based on this basic problem of beta-expansion, we discuss the length of a cylinder and the distribution of cylinders with regular lengths.The thesis is organized as follows:in the introductory chapter we mainly introduce the background and current situations of the research, and then cite some well known results concerning beta-expansion.In the second chapter, we cite the definitions, elementary properties and relevant con-clusions for later use.The third chapter constitutes the main part of this thesis:a criterion of full cylinders their distribution.The proofs are given in the last chapter. As an application to the metrical theory of beta-expansion, we determine the Hausdorff dimension of the set: where{zn}n≥1is a sequence of real numbers in [0,1],f:[0,1]â†'R+is a positive continuous function and Snf(x) denotes the ergodic sum f(x)+…+f(Tβnx).