BADLY APPROXIMABLE AND NONRECURRENT SETS FOR EXPANDING MARKOV MAPS

Na Yuan (), Bing Li () and Min Wu
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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

FRACTALS (fractals), 2021, vol. 29, issue 06, 1-17

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.

Keywords: Markov Map; Hausdorff Dimension; Nonrecurrent Set (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X2150170X

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