2018/03/20 by Charlene Kalle, Derong Kong, Kalle, Charlene +5 · 4 citations
Mathematics · #11A63 #11K55 #26A30 #28D05 #37B10 #37E05 #37E15 #68R15 #Advanced Topology and Set Theory #BETA (programming language) #Combinatorics #Dimension (graph theory) #Discrete mathematics #Dynamical Systems (math.DS) #FOS: Mathematics #Hausdorff dimension #Hausdorff space #Lebesgue integration #Lebesgue measure #Mathematical Dynamics and Fractals #Mathematics #math.DS #msc:11A63 #msc:11K55 #msc:26A30 #msc:28D05 #msc:37B10 #msc:37E05 #msc:37E15 #msc:68R15
paper · pdf · doi:10.48550/arxiv.1803.07338
published in arXiv (Cornell University) (Cornell University) · 32 pages, 4 figures
arxiv created 2018/03/20 · openalex publication_date 2018/03/20 · arxiv updated 2018/03/21 · openalex created_date 2018/03/29 · openalex updated_date 2026/08/08
For β∈(1,2] the β-transformation Tβ: [0,1) → [0,1) is defined by Tβ( x) = βx \pmod 1. For t∈[0, 1) let Kβ(t) be the survivor set of Tβ with hole (0,t) given by Kβ(t):=\x∈[0, 1): Tβn(x)\not ∈ (0, t) \textrm for all n≥ 0\. In this paper we characterise the bifurcation set Eβ of all parameters t∈[0,1) for which the set valued function t↦ Kβ(t) is not locally constant. We show that Eβ is a Lebesgue null set of full Hausdorff dimension for all β∈(1,2). We prove that for Lebesgue almost every β∈(1,2) the bifurcation set Eβ contains both infinitely many isolated and accumulation points arbitrarily close to zero. On the other hand, we show that the set of β∈(1,2) for which Eβ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for E2, the bifurcation set of the doubling map. Finally, we give for each β∈ (1,2) a lower and upper bound for the value τβ, such that the Hausdorff dimension of Kβ(t) is positive if and only if t< τβ. We show that τβ≤ 1-\frac1β for all β∈ (1,2).