The Diophantine Approximation Problem In One-dimensional Expanding Markov And Affine Systems

The classic Diophantine approximation is a branch of number theory that mainly studies problem of approximating irrational numbers with rational numbers.The dynamical Diophantine approximation studies the related problem of the approximation of a given point y ∈ X by the orbits ｛Tnx｝n≥0 of a point x ∈ X in the dynamical system(X,T).The set of points x∈X whose orbits can approximate y well is called a well-approximable set,also known as a shrinking target set.Correspondingly,the set of points whose orbits cannot approximate y well is called a badly approximable set.When considering the approximation of the initial point x by its own orbit,the set is called a recurrence set.There has been a large amount of research on shrinking target sets and recurrence sets,such as Hill and Velani(1995),and Boshernitzan(1993).In this thesis,we investigate the above problems for one-dimensional expanding Markov systems and affine systems,and obtain results in the following three aspects.Firstly,research on the badly approximable sets and non-recurrent sets in one dimension expanding Markov systems.In the expanding Markov system([0,1],T),｛yn｝n≥0(?)[0,1]is a given sequence,we prove that the badly approximable set{x∈[0,1]:(?)|Tn(x)-yn|>0},has full Hausdorff dimension.This result can be applied to Gauss maps and Lüroth maps,corresponding to dynamical systems consisting of countably infinitely many symbols.As another application of the above result,we prove that the Hausdorff dimension of the non-recurrent set in the one-dimensional Markov system is also full.This is a supplement to the result of Boshernitzan(1993),who proved that almost all points are recurrent.In addition,we prove that non-resurrent set is dense in the interval[0,1].Secondly,research on the well-approximated sets and the recurrent sets under Lipschitz functions in one-dimensional expanding Markov systems.Let X (?)[0,1],in the expanding Markov system(X,T),it is proved that the set R(ψ,{gn})=｛x∈X:|Tnx-gn(x)|<ψ(n)for infinitely many n∈ N}has full Hausdorff dimension,where ψ:N→R+ and {gn}n≥1 is a sequence of functions defined on the set X with a uniform Lipschitz constant.This result can not only be applied to systems of interval map with a finite and countable number of branches but also to dynamical systems defined on fractals,such as cookie-cutter dynamical systems.Thirdly,research on the reurrence sets in affine systems.We studied the recurrence sets of a class of Bedford-McMullen carpets K,where the inverse of the iterated function system generating K is the transformation T in the dynamical system.When the Box dimension and Hausdorff dimension of K are equal,the Hausdorff dimension of the recurrence set is obtained.The main difficulty of this problem lies in the lack of effective tools similar to the Mass transference principle when K cannot be represented as the product of two Cantor-type sets.Therefore,the lower bound problem for the dimension of the recurrence set in K requires a considerable amount of meticulous calculation.This result can not only solve the problem of recurrence sets in the product form of two Cantor-type sets but also some cases that are not in the product form.