Quantitative Recurrence Properties in Some Irregular Sets for Beta Dynamical Systems

Yuanyang Chang () and Wenna Liu
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Yuanyang Chang: School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070, China
Wenna Liu: School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070, China

Mathematics, 2025, vol. 13, issue 11, 1-22

Abstract: Let β > 1 be a real number and T β x = β x ( m o d 1 ) . This paper is concerned with the quantitative recurrence properties of the system ( [ 0 , 1 ] , T β ) in some (refined) irregular sets. Specifically, let α 1 , α 2 > 0 and ψ : N → ( 0 , 1 ) be a positive function; we define the set E α 1 , α 2 β = x ∈ [ 0 , 1 ) : lim inf n → ∞ 1 n S n ( x , β ) = α 1 , lim sup n → ∞ 1 n S n ( x , β ) = α 2 , and calculate the Hausdorff dimension of the set E α 1 , α 2 β ( ψ ) : = x ∈ E α 1 , α 2 β : | T β n x − x | < ψ ( n ) i . m . n ∈ N , where i . m . stands for infinitely many. Consequently, the Hausdorff dimension of the set E ^ β ( ψ ) = x ∈ [ 0 , 1 ) : lim n → ∞ 1 n S n ( x , β ) does not exist , | T β n x − x | < ψ ( n ) i . m . n ∈ N is also determined.

Keywords: β -expansion; irregular sets; quantitative recurrence; Hausdorff dimension (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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