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Relative bifurcation sets and the local dimension of univoque bases

Part of: Elementary number theory Topological dynamics Classical measure theory

Published online by Cambridge University Press:  26 May 2020

PIETER ALLAART  and
DERONG KONG
PIETER ALLAART
Affiliation:
Mathematics Department, University of North Texas, 1155 Union Cir #311430, Denton, TX76203-5017, USA (e-mail: allaart@unt.edu)
DERONG KONG
Affiliation:
College of Mathematics and Statistics, Chongqing University, 401331, Chongqing, PR China (e-mail: derongkong@126.com)
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Abstract

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Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$. The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit

$$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$
for all $q\in (1,M+1)$. We show in particular that $f(q)>0$ if and only if $q\in \overline{\mathscr{U}}\backslash \mathscr{C}$, where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al [On the bifurcation set of unique expansions. Acta Arith. 188 (2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets. Adv. Math.308 (2017), 575–598] on strongly univoque sets.

Keywords

univoque basesHausdorff dimensionbifurcation setrelative entropy plateaustrongly univoque set

MSC classification

Primary: 11A63: Radix representation; digital problems
Secondary: 37B10: Symbolic dynamics 28A80: Fractals
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 8 , August 2021 , pp. 2241 - 2273
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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