Some Studies On ?-expansions

The theory of real numbers is a central problem in number theory, which is close-ly related to many other fields such as metrical number theory, dynamical system, fractal geometry and combinatorics on words. In this thesis, we study some problems on the ? expansions of real numbers. We consider the comparison of the bases under which some number has a periodic unique expansion, and characterize the real numbers with exactly countably many expansions. The thesis is organized as follows.In the first chapter, we introduce the background of the dynamical system and the expansions of real numbers. In the second chapter, we mainly present the definitions and properties of the ? expansions, as well as the preliminaries. In the next two chapters, the two issues are discussed in details.In the third chapter, we study, for any m, n ? N, the set of bases ? such that there exists x? (0,m/?-1) whose ?-expansion with the digits 0,1,..., m is unique and periodic with the smallest period n, more precisely,where ??,m is the set of the unique expansions,Pn denotes the set of the pure periodic sequences with the smallest period n. Allouche, Clarke and Sidorov showed in the two digits case (that is m=1) that Pn,1=(?n,2) for some ?n ?(1,2). Furthermore, they proved that ?k>?l if and only if k comes before l in the sense of the Sharkheovskii ordering. Answering an open question posed by Baker, we study the case of the general digit-set. We remark that in the case m being odd, the methods and techniques in can be directly applied to obtain a similar results. But the case with m being even becomes much more difficult and involved. By lifting up the problem to a bigger digit-set, and then projecting down to the original digit-set, we tackle this problem.In the forth chapter, we characterize the set of the elements which have exactly count-ably many expansions in the generalized golden rations. The generalized golden ratio is the threshold value for the base such that there exists some nontrivial real number with at most countably many expansions with the given digits. Sidorov and Vershik showed that with the digits 0,1 and in base G=(?)/2 the numbers x=nG mod1 have countably many expansions for any n ? Z, while the other elements of (0,1/G-1) have uncountably many expansions.We study the general case with the digit set {0,1,…,m} and the base the generalized golden ratio ?=y(m).If m=2k+1,g(m)=k+1+(?)/2,the num-bers x=p?+q/(k+1)n?(0,m/?-1)(n,p,q?Z) have countably infinitely many expansions?other elements of(0,m/?-1) have uncountably many expansions;if m=2k,g(m)=k+1,the numbers with countably many expansions are p/(k+1)n?(0,2)(n,p?N?{0})In the final chapter,we summarize the main results of thisis,esis,and then pose some questions for further study.