Quantitative Poincar recurrence in continued fraction dynamical system

作     者：PENG Li TAN Bo WANG BaoWei 

作者机构：School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhan 430074China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2012年第55卷第1期

页      面：131-140页

核心收录：

学科分类：090603[农学-临床兽医学] 07[理学] 09[农学] 0906[农学-兽医学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by National Natural Science Foundation of China (Grant Nos.10631040 10901066) 

主　　题：recurrence continued fraction Hausdorff dimension 

摘      要：Let T ： X → X be a transformation. For any x C [0, 1） and r 〉 O, the recurrence time Tr（x） of x under T in its r-neighborhood is defined as Tr（X） = inf{k ≥ 1： d（Tk（x）,x） 〈 r}.For 0 ≤ α ≤ β ∞ co, let E（α,β） be the set of points with prescribed recurrence time as follows E（α,β）={x∈X：lim inf r→0 logTr（x）/-logr=α,lim sup r→0 logTr（x）/-logr=β}.In this note, we consider the Gauss transformation T on [0, 1）, and determine the size of E（α,β）by showing that dimH E（α,β） = 1 no matter what a and/~ are. This can be compared with Feng and Wu＇s result [Nonlinearity, 14 （2001）, 81-85] on the symbolic space.