{"title":"Bifurcation sets arising from non-integer base expansions","authors":"P. Allaart, S. Baker, D. Kong","doi":"10.4171/jfg/79","DOIUrl":null,"url":null,"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 \n\\[ \nx=\\frac{x_1}{q}+\\frac{x_2}{q^2}+\\frac{x_3}{q^3}+\\cdots. \n\\] \nDenote by $\\mathbf U_q$ the set of corresponding sequences of all points in $\\mathcal U_q$. \nIt 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 \n\\[ \n\\mathcal B:=\\{q\\in(1,M+1]: H(p)\\ne H(q)\\textrm{ for any }p\\ne q\\}. \n\\] 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 $x^5-x^4-x^3-2x^2+x+1=0$.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":" ","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2017-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jfg/79","citationCount":"12","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Fractal Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/jfg/79","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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 $x^5-x^4-x^3-2x^2+x+1=0$.