Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6418255 | Journal of Mathematical Analysis and Applications | 2014 | 15 Pages |
Abstract
For any real number β>1, we denote by Tβ the β-transformation on the unit interval [0,1] given by Tβ(x)=βxââβxâ, where âξâ denotes the integer part of ξ. We consider the size of the set of β>1 where the orbit of 1 under Tβ ultimately has a positive distance from a given point in [0,1]. For any x0â[0,1] and any (β0,β1)â(1,â), we obtain the set of βâ(β0,β1) such that x0 is not an accumulation point of the orbit of 1 under Tβ with full Hausdorff dimension. Specifically,dimH{βâ(β0,β1):liminfnââ|Tβn1âx0|>0}=1. This is a generalization of the result of Schmeling for the case where x0=0 and (β0,β1)=(1,â).
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Chun-Yun Cao,