Research On Some Fractal Sets In Theexpansion Of Numbers

In the study of digit expansions and related dynamical systems,the sets defined by some constraints on digits of expansions and many invariant sets of dynamical systems are usually fractals.It is a hot topic to study the structures and the Hausdorff dimensions of these fractals in number theory and dynamical systems.Schmidt's game is an important tool for studying the countable intersection of fractal sets.It has been widely used in the related study by researchers of different countries in recent years.The expansion of real numbers with negative bases,which was introduced for about ten years ago,is a generalization of the classic expansion with positive bases.Compared with ?-expansion and ?-transformation,(-?)-expansion and the related dynamical system have different combinatorial structures and topological properties.In the thesis,we mainly study two problems.The one is the property of Schmidt's(?,?)-game when the parameters a and b are changed,and the other is the Hausdorff dimension of the set of points with non-dense orbits under the(-?)-transformation.The thesis is organized as follows.The first chapter is the introduction.It contains the background of Schmidt's game and the expansions of real numbers with non-integer bases.Firstly,we introduce some applications of Schmidt's game in Diophantine approximation and many problems of orbits of certain dynamical systems in number theory.Some results of Schmidt's(?,?)-game about the parameters a and b are also introduced.Then we introduce the background and known results of expansions with non-integer bases.Finally,we introduce the two problems studied in the thesis.The second chapter is the preliminary,which consists of some definitions and basic properties in fractal geometry,dynamic systems and digital expansions.It contains the definition and basic properties of Hausdorff measure and dimension,symbolic space,several different expansions of numbers,Schmidt's(?,?)-game and the winning sets.In the third chapter,we study the property of Schmidt's(?,?)-game when parameters a and b are changed.In Schmidt's book about Diophantine approximation,Schmidt asked the question whether the(?,?)-winning set is still winning when the parameter a decreases.This question was answered in the negative by Freiling.We ask the question whether the set still remains winning when the parameter b increases and give an answer to it.In the fourth chapter,we study the Hausdorff dimension of the set of points withnon-dense orbits under the(-?)-transformation and prove that the set is of full dimension.In this case of ?-transformation,similar sets are always of zero measure and full dimension.But when b is less than the reciprocal of the golden ratio,the set is not of zero measure for the(-?)-transformation.In the fifth chapter,we summarize the thesis,and introduce some problems for the further research.