Problems Of The Distribution Of Orbits In Several Types Of Expansions Of Real Numbers

In this thesis,we discuss some questions about the expansions of real numbers.Firstly,We study the necessary and sufficient condition under which(-?)-shift has the specification property in the sense of Bowen,then obtain the Hausdorff dimension of the set of ? satisfying this condition.Secondly,we define exceptional sets about the maximal length of the consecutive 0 digits of ?-expansions,then obtain the Hausdorff dimension of the exceptional sets.Finally,we investigate the relative growth rate of the partial quotients of Engel continued fractions.This thesis is consist of five chapters.In chapter 1,we introduce the history and background of fractal geometry.Then we give the introduction of the origin and development of main results considered in this paper.We also list the main results briefly.In chapter 2,we recall the basic facts of Hausdorff dimension,the expansions of real numbers and dynamical system.In chapter 3,we show the necessary and sufficient condition under which(-?)-shift has the specification property is that ??(?)+1/2 and sup {T-?2n-1(1):n? 1}<1.At the same time,we obtain the set of ? for which(-?)-shift has the specification property is full in the sense of dimension.In chapter 4,we consider the maximal length of the consecutive zero digits of ?-expansions in the sense of dimension.Then the Hausdorff dimension of the relative exceptional set of ? is full.In chapter 5,it is proved that for Lebesgue almost all x ?(0,1),the relative growth speed of partial quotients of the Engel continued fractions is exponential growth rate.Furthermore,related exceptional sets consisting of points fail holding this speed is full in the sense of dimension.