2016/04/06 by Lixuan Zheng, Min Wu, Zheng, Lixuan +3
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #advanced mathematical theories #math.DS
paper · pdf · doi:10.48550/arxiv.1604.01470
13 pages, 5 main results
openalex publication_date 2016/04/06 · openalex created_date 2016/06/24 · arxiv created 2016/12/15 · arxiv updated 2016/12/16 · openalex updated_date 2026/07/28
Let β> 1 be a real number and (ε1(x, β), ε2(x, β), …) be the β-expansion of a point x ∈ (0, 1]. For all x ∈ (0,1], let A(D(x)) be the set of accumulation points of (-logβ|In(x)|)/(n) as n → ∞, where |In(x)| is the length of the basic interval of order n containing x ∈ (0, 1]. In this paper, we prove that A(D(x)) is always a closed interval for any x ∈ (0,1]. Furthermore, if λ(β)>0, the extremely irregular set containing points x ∈ [0, 1] whose upper limit of (-logβ|In(x)|)/(n) equals to 1+ł(β) is residual, where 1+ł(β) is a constant depending on β. As a consequence, the irregular set with x∈ [0, 1] whose limit of (-logβ|In(x)|)/(n) does not exist is residual for every λ(β)>0.