The β-transformation with a hole

2014/12/31 by Lyndsey Clark
Computer Science · Mathematics · #Holomorphic and Operator Theory #Integer (computer science) #Mathematical Dynamics and Fractals #Period (music) #Period-doubling bifurcation #Periodic point #Point (geometry) #Set (abstract data type) #math.DS #msc:28D05 #msc:37B10 #msc:68R15 #semigroups and automata theory

paper · pdf · doi:10.3934/dcds.2016.36.1249

published as Discrete and Continuous Dynamical Systems A, Volume 36, Issue 3, pages 1249-1269, March 2016 · 20 pages, 8 figures. Updated version has added examples and a small mistake in Lemma 3.4 has been fixed

openalex publication_date 2015/08/28 · arxiv created 2015/09/18 · arxiv updated 2015/09/21 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general β-transformation. Let β ∈ (1,2) and consider the β-transformation T_β(x)=β x mod 1. Let J_β(a,b) := \ x ∈ (0,1) : T_βⁿ(x) ∉ (a,b) for all n ≥ 0 \. An integer n is bad for (a,b) if every periodic point of period n for T_β intersects (a,b). Denote the set of all bad n for (a,b) by B_β(a,b). In this paper we completely describe the following sets:D₀(β) = \ (a,b) ∈ [0,1)² : J(a,b) ≠ ∅ \,
D₁(β) = \ (a,b) ∈ [0,1)² : J(a,b) is uncountable \,
D₂(β) = \ (a,b) ∈ [0,1)² : B_β(a,b) is finite \.

Citations

• On theβ-expansions of real numbers
• Representations for real numbers and their ergodic properties
• The doubling map with asymmetrical holes