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- W3137561271 abstract "It was conjectured by Furstenberg that for any x∈[0,1]∖Q, dimH{2nx(mod1):n≥1}¯+dimH{3nx(mod1):n≥1}¯≥1. Where dimH denotes the Hausdorff dimension and A¯ denotes the closure of a set A. Can we obtain analogous dimension formula when considering the point under the β-expansions of different bases? In this paper, we are aiming at giving explicit non-normal numbers for which the above dimensional formula holds. We illustrate our result as follows: For any β1>1 and β2>1,there exists a Cantor set E composed of real numbers in the unit interval such that 1. Each x∈E is not β1-normal; 2. For each x∈E, {Tβ1nx:n≥1}¯=[0,1] and {Tβ2nx:n≥1}¯=[0,1]; 3. dimHE=1. Where Tβ denotes the β-transformation." @default.
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- W3137561271 date "2021-05-01" @default.
- W3137561271 modified "2023-10-11" @default.
- W3137561271 title "The points with dense orbit under the <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.svg><mml:mi>β</mml:mi></mml:math>-expansions of different bases" @default.
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- W3137561271 doi "https://doi.org/10.1016/j.chaos.2021.110840" @default.
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