The Distribution Properties Of Orbits In Binary Dynamical System

The study of the Diophantine properties of the orbits in dynamical system has extended the classical Diophantine approximation thought and become an important subject in the study of dynamical systems and Diophantine approximation.This article mainly discusses the distribution properties of orbits in binary dynamical system.The Lebesgue measure and Hausdorff measure of the related set is calculated.This paper is divided into five chapters.The first chapter is the introduction,which introduces the background and present research situation of the research problems in this paper.The second chapter is the preliminary knowledge,mainly including the relevant definitions and theorems that will be used later in the article.The third and fourth chapters are the main part of this paper.Let ψ:N→R+ be a positive function,a transformation T:[0,1]→[0,1]is defined by T(x)=2x(modl)for all x ∈[0,1]Let E(ψ)={(x,y)∈[0,1]x[0,1]:|Tnx-y|<ψ(n)for infinitely many n∈N｝.In the third chapter,it is proved that the Lebesgue measure of the set E(ψ)satisfying the zero-one laws.In the fourth chapter,slicing lemma together with the mass transference principle allows us to transfer Lebesgue measure theoretic statements for limsup set E(ψ)to Hausdorff measure theoretic statements.Let f be a dimension function such that f(x)/x is decreasing and g(x)=xf(x),it is proved that the Hausdorff measure of the set E(ψ)satisfyingIn the fifth chapter,we give the summary of this paper and prospect of the following research work.