Hausdorff dimension of univoque sets and Devil's staircase

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• 作者：Vilmos Komornik^a ; ^{komornik@math.unistra.fr" class="auth_mail" title="E-mail the corresponding author} ; Derong Kong^b ; ^{derongkong@126.com" class="auth_mail" title="E-mail the corresponding author} ; Wenxia Li^c ; ^{wxli@math.ecnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 11A63 ; secondary ; 10K50 ; 11K55 ; 37B10 ; 28A80
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：165-196
• 全文大小：500 K

文摘

We fix a positive integer M  , and we consider expansions in arbitrary real bases q>1$q>1$ over the alphabet {0,1,…,M}$\{0,1,\dots ,M\}$. We denote by U_q${\mathcal{U}}_q$ the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension D(q)$D(q)$ of U_q${\mathcal{U}}_q$ for each q∈(1,∞)$q\in (1,\infty )$. Furthermore, we prove that the dimension function D:(1,∞)→[0,1]$D:(1,\infty )\to [0,1]$ is continuous, and has bounded variation. Moreover, it has a Devil's staircase behavior in (q^′,∞)$(q^{\prime },\infty )$, where q^′$q^{\prime }$ denotes the Komornik–Loreti constant: although D(q)>D(q^′)$D(q)>D(q^{\prime })$ for all q>q^′$q>q^{\prime }$, we have D^′<0$D^{\prime }<0$ a.e. in (q^′,∞)$(q^{\prime },\infty )$. During the proofs we improve and generalize a theorem of Erdős et al. on the existence of large blocks of zeros in β-expansions, and we determine for all M   the Lebesgue measure and the Hausdorff dimension of the set U$\mathcal{U}$ of bases in which x=1$x=1$ has a unique expansion.