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Badly Approximable And Nonrecurrent Sets For Expanding Markov Maps

Author

Listed:
  • NA YUAN

    (Department of Mathematics, South China University of Technology, Guangzhou 510640, P. R. China)

  • BING LI

    (Department of Mathematics, South China University of Technology, Guangzhou 510640, P. R. China)

  • MIN WU

    (Department of Mathematics, South China University of Technology, Guangzhou 510640, P. R. China)

Abstract

We consider the asymptotic behaviors of the orbits of an expanding Markov system ([0, 1],f), and prove that the badly approximable set {x ∈ [0, 1) :liminfn→∞|fn(x) − y n| > 0}, is of full Hausdorff dimension for any given sequence {yn}n≥0 ⊂ [0, 1]. Consequently, the Hansdorff dimension of the set of nonrecurrent points in the sense that {x ∈ [0, 1] :liminfn→∞|fn(x) − x| > 0} is also full. The results can be applied to β-transformations, Gauss maps and Lüroth maps, etc.

Suggested Citation

  • Na Yuan & Bing Li & Min Wu, 2021. "Badly Approximable And Nonrecurrent Sets For Expanding Markov Maps," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(06), pages 1-17, September.
    • Handle: RePEc:wsi:fracta:v:29:y:2021:i:06:n:s0218348x2150170x
    DOI: 10.1142/S0218348X2150170X
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