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The points with dense orbit under the β-expansions of different bases

Author

Listed:
  • Wang, Wen-Ya
  • Chen, Hui-Qin
  • Guo, Zhong-Kai

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.

Suggested Citation

  • Wang, Wen-Ya & Chen, Hui-Qin & Guo, Zhong-Kai, 2021. "The points with dense orbit under the β-expansions of different bases," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    • Handle: RePEc:eee:chsofr:v:146:y:2021:i:c:s0960077921001934
    DOI: 10.1016/j.chaos.2021.110840
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