Quantitative Poincar recurrence in continued fraction dynamical system

Let T:X → X be a transformation.For any x ∈[0,1) and r > 0,the recurrence time τr(x) of x under T in its r-neighborhood is defined as τr(x)=inf k 1:d (Tk(x),x) < r.For 0 αβ∞,let E(α,β) be the set of points with prescribed recurrence time as follows E (α,β)=x ∈ X:lim r→0 inf[log τr(x)/-log r]=α,lim r→0 sup[log τr(x)/-log r]=β.In this note,we consider the Gauss transformation T on [0,1),and determine the size of E (α,β) by showing that dim H E (α,β)=1 no matter what α and β are.This can be compared with Feng and Wu's result [Nonlinearity,14 (2001),81-85] on the symbolic space.

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.