1 results
Relative bifurcation sets and the local dimension of univoque bases
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 8 / August 2021
- Published online by Cambridge University Press:
- 26 May 2020, pp. 2241-2273
- Print publication:
- August 2021
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- Article
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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 for all
$$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$
$q\in (1,M+1)$. 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.
