Multifractal analysis of the divergence points of Birkhoff averages in beta-dynamical systems

2015/12/31 by Yuanhong Chen, Zhenliang Zhang, Chen, Yuanhong +3
Mathematics · #11K55 #28A80 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.DS #math.NT #msc:11K55 #msc:28A80

paper · pdf · doi:10.48550/arxiv.1512.09205

non

arxiv created 2015/12/31 · openalex publication_date 2015/12/31 · arxiv updated 2016/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of β-expansions. More precisely, let ([0,1),T_β) be the β-dynamical system for a general β>1 and ψ:[0,1]↦ℝ be a continuous function. Denote by \textsfA(ψ,x) all the accumulation points of \(1)/(n)∑_j=0^n-1ψ(T^jx): n≥ 1\. The Hausdorff dimensions of the sets \x:\textsfA(ψ,x)⊃[a,b]\, \x:\textsfA(ψ,x)=[a,b]\, \x:\textsfA(ψ,x)⊂[a,b]\ i.e., the points for which the Birkhoff averages of ψ do not exist but behave in a certain prescribed way, are determined completely for any continuous function ψ.

Citations

• Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension
• Irregular sets, the 𝛽-transformation and the almost specification property
• Representations for real numbers and their ergodic properties

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