Bifurcation sets arising from non-integer base expansions

IF 1.1 4区 数学 Q1 MATHEMATICS
P. Allaart, S. Baker, D. Kong
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引用次数: 12

Abstract

Given a positive integer $M$ and $q\in(1,M+1]$, let $\mathcal U_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion: there exists a unique sequence $(x_i)=x_1x_2\ldots$ with each $x_i\in\{0,1,\ldots, M\}$ such that \[ x=\frac{x_1}{q}+\frac{x_2}{q^2}+\frac{x_3}{q^3}+\cdots. \] Denote by $\mathbf U_q$ the set of corresponding sequences of all points in $\mathcal U_q$. It is well-known that the function $H: q\mapsto h(\mathbf U_q)$ is a Devil's staircase, where $h(\mathbf U_q)$ denotes the topological entropy of $\mathbf U_q$. In this paper we {give several characterizations of} the bifurcation set \[ \mathcal B:=\{q\in(1,M+1]: H(p)\ne H(q)\textrm{ for any }p\ne q\}. \] Note that $\mathcal B$ is contained in the set $\mathcal{U}^R$ of bases $q\in(1,M+1]$ such that $1\in\mathcal U_q$. By using a transversality technique we also calculate the Hausdorff dimension of the difference $\mathcal B\backslash\mathcal{U}^R$. Interestingly this quantity is always strictly between $0$ and $1$. When $M=1$ the Hausdorff dimension of $\mathcal B\backslash\mathcal{U}^R$ is $\frac{\log 2}{3\log \lambda^*}\approx 0.368699$, where $\lambda^*$ is the unique root in $(1, 2)$ of the equation $x^5-x^4-x^3-2x^2+x+1=0$.
由非整数基展开引起的分歧集
给定一个正整数$M$和$q\in(1,M+1]$,设$\mathcal U_q$是[0,M/(q-1)]$中的$x\in的集合,具有唯一的$q$-展开式:存在一个唯一序列$(x_i)=x_1x2\ldots$,每个$x_i\in \{0,1,\ldots,M\}$,使得\[x=\frac{x_1}bf U_ q$中所有点的对应序列的集合。众所周知,函数$H:q\mapsto H(\mathbf U_q)$是魔鬼楼梯,其中$H(\math bf U_q)$表示$\mathbfU_q$的拓扑熵。在本文中,我们{给出了}分支集{[\mathcal B:=\{q\In(1,M+1]:H(p)\ne H(q)\textrm{对于任何}p\ne q\}的几个特征注意$\mathcal B$包含在基$q\in(1,M+1]$的集合$\mathical{U}^R$中,使得$1\in\mathcal U_q$。通过使用横截性技术,我们还计算差值$\mathcalB\反斜杠\mathcal{U}^R$的Hausdorff维数。有趣的是,这个量总是严格地在$0$和$1$之间。当$M=1$时,$\mathal B\反斜线\mathcal{U}的Hausdoff维数^R$是$\frac{\log 2}{3\log\lambda ^*}\约0.368699$,其中$\lambda ^*$是方程$x^5-x^4-x^3-2x^2+x+1=0$的$(1,2)$中的唯一根。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.50
自引率
0.00%
发文量
9
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