The Hausdorff Dimensions Of Recurrent Sets In The Field Of Formal Laurent Series

The study of recurrence problem in dynamical systems can be dated back to the work of famous French mathematician Poincaré.When he studied the N-body problem,he com-pletely put forward the concept of invariant integral and used invariant integral as a tool to prove the classical Poincaré recurrence theorem.On this basis,we study the recurrent problem in continued fraction dynamical system over the field of formal series and mainly consider the case of recurrence at polynomial speed.This paper is divided into four chapters.The first two chapters mainly introduces the background,progress and preliminary knowledge of relevant research.The third chapter is the main part of this paper.Let Fq be a finite field of q elements,(?)Let x?(?)and[a1(x),a2(x),a3(x),...]be the continued fraction expansion of x.Denote by(?)n(x)the admissible cylinder of order n containing x and by ?n(x)the first return time of x?(?)into(?)n(x),Then?n(x)=inf{k?1:Tk(x)?(?)n(x)}.In this paper,we mainly study the Hausdorff dimension of the sets(?)and we show that E(?,?)is of full Hausdorff dimension for any 1??????.In the last Chapter,We give some reflections about the results of this paper.