On the bifurcation set of unique expansions
On the bifurcation set of unique expansions
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关于唯一展开式的分叉集
DOI：
10.4064/aa171212-11-7
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发表时间：
2016-12
期刊：
影响因子：
0.7
通讯作者：
Lu Fan
中科院分区：
文献类型：
--
作者：
Kalle Charlene;Kong Derong;Li Wenxia;Lu Fan
Given a positive integer $M$, for $q\in(1, M+1]$ let ${\mathcal{U}}_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion with the digit set $\{0, 1,\ldots, M\}$, and let $\mathbf{U}_q$ be the set of corresponding $q$-expansions. Recently, Komornik et al.~(Adv. Math., 2017) showed that the topological entropy function $H: q \mapsto h_{top}(\mathbf{U}_q)$ is a Devil's staircase in $(1, M+1]$. Let $\mathcal{B}$ be the bifurcation set of $H$ defined by
\[
\mathcal{B}=\{q\in(1, M+1]: H(p)\ne H(q)\quad\textrm{for any}\quad p\ne q\}.
\]
In this paper we analyze the fractal properties of $\mathcal{B}$, and show that for any $q\in \mathcal{B}$,
\[
\lim_{\delta\rightarrow 0} \dim_H(\mathcal{B}\cap(q-\delta, q+\delta))=\dim_H\mathcal{U}_q,
\] where $\dim_H$ denotes the Hausdorff dimension. Moreover, when $q\in\mathcal{B}$ the univoque set $\mathcal{U}_q$ is dimensionally homogeneous, i.e., $
\dim_H(\mathcal{U}_q\cap V)=\dim_H\mathcal{U}_q $ for any open set $V$ that intersect $\mathcal{U}_q$.
As an application we obtain a dimensional spectrum result for the set $\mathcal{U}$ containing all bases $q\in(1, M+1]$ such that $1$ admits a unique $q$-expansion. In particular, we prove that for any $t>1$ we have \[
\dim_H(\mathcal{U}\cap(1, t])=\max_{ q\le t}\dim_H\mathcal{U}_q. \] We also consider the variations of the sets $\mathcal{U}=\mathcal{U}(M)$ when $M$ changes.
Given a positive integer $M$, for $q\in(1, M+1]$ let ${\mathcal{U}}_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion with the digit set $\{0, 1,\ldots, M\}$, and let $\mathbf{U}_q$ be the set of corresponding $q$-expansions. Recently, Komornik et al.~(Adv. Math., 2017) showed that the topological entropy function $H: q \mapsto h_{top}(\mathbf{U}_q)$ is a Devil's staircase in $(1, M+1]$. Let $\mathcal{B}$ be the bifurcation set of $H$ defined by
\[
\mathcal{B}=\{q\in(1, M+1]: H(p)\ne H(q)\quad\textrm{for any}\quad p\ne q\}.
\]
In this paper we analyze the fractal properties of $\mathcal{B}$, and show that for any $q\in \mathcal{B}$,
\[
\lim_{\delta\rightarrow 0} \dim_H(\mathcal{B}\cap(q-\delta, q+\delta))=\dim_H\mathcal{U}_q,
\] where $\dim_H$ denotes the Hausdorff dimension. Moreover, when $q\in\mathcal{B}$ the univoque set $\mathcal{U}_q$ is dimensionally homogeneous, i.e., $
\dim_H(\mathcal{U}_q\cap V)=\dim_H\mathcal{U}_q $ for any open set $V$ that intersect $\mathcal{U}_q$.
As an application we obtain a dimensional spectrum result for the set $\mathcal{U}$ containing all bases $q\in(1, M+1]$ such that $1$ admits a unique $q$-expansion. In particular, we prove that for any $t>1$ we have \[
\dim_H(\mathcal{U}\cap(1, t])=\max_{ q\le t}\dim_H\mathcal{U}_q. \] We also consider the variations of the sets $\mathcal{U}=\mathcal{U}(M)$ when $M$ changes.
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影响因子：
0.9
作者：
Carlo Carminati;G. Tiozzo
通讯作者：
Carlo Carminati;G. Tiozzo
影响因子：
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作者：
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DOI：
10.1016/j.bpa.2005.07.009
发表时间：
2006-03-01
影响因子：
4.8
作者：
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通讯作者：
Bein, Berthold
DOI：
10.7551/mitpress/11924.003.0011
发表时间：
2022-01
期刊：
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影响因子：
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作者：
C. Tsallis
通讯作者：
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影响因子：
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作者：
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