Critical values for the $β$-transformation with a hole at $0$
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arXiv
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2021
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| _version_ | 1866915800435130368 |
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| author | Allaart, Pieter Kong, Derong |
| author_facet | Allaart, Pieter Kong, Derong |
| contents | Given $β\in(1,2]$, let $T_β$ be the $β$-transformation on the unit circle $[0,1)$ such that $T_β(x)=βx\pmod 1$. For each $t\in[0,1)$ let $K_β(t)$ be the survivor set consisting of all $x\in[0,1)$ whose orbit $\{T^n_β(x): n\ge 0\}$ never hits the open interval $(0,t)$. Kalle et al. proved in [Ergodic Theory Dynam. Systems, 40 (9): 2482--2514, 2020] that the Hausdorff dimension function $t\mapsto\dim_H K_β(t)$ is a non-increasing Devil's staircase. So there exists a critical value $τ(β)$ such that $\dim_H K_β(t)>0$ if and only if $t<τ(β)$. In this paper we determine the critical value $τ(β)$ for all $β\in(1,2]$, answering a question of Kalle et al. (2020). For example, we find that for the Komornik-Loreti constant $β\approx 1.78723$ we have $τ(β)=(2-β)/(β-1)$. Furthermore, we show that (i) the function $τ: β\mapstoτ(β)$ is left continuous on $(1,2]$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $τ$ has no downward jumps, with $τ(1+)=0$ and $τ(2)=1/2$; and (iii) there exists an open set $O\subset(1,2]$, whose complement $(1,2]\setminus O$ has zero Hausdorff dimension, such that $τ$ is real-analytic, convex and strictly decreasing on each connected component of $O$. Our strategy to find the critical value $τ(β)$ depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_10012 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Critical values for the $β$-transformation with a hole at $0$ Allaart, Pieter Kong, Derong Dynamical Systems Classical Analysis and ODEs Combinatorics Number Theory Given $β\in(1,2]$, let $T_β$ be the $β$-transformation on the unit circle $[0,1)$ such that $T_β(x)=βx\pmod 1$. For each $t\in[0,1)$ let $K_β(t)$ be the survivor set consisting of all $x\in[0,1)$ whose orbit $\{T^n_β(x): n\ge 0\}$ never hits the open interval $(0,t)$. Kalle et al. proved in [Ergodic Theory Dynam. Systems, 40 (9): 2482--2514, 2020] that the Hausdorff dimension function $t\mapsto\dim_H K_β(t)$ is a non-increasing Devil's staircase. So there exists a critical value $τ(β)$ such that $\dim_H K_β(t)>0$ if and only if $t<τ(β)$. In this paper we determine the critical value $τ(β)$ for all $β\in(1,2]$, answering a question of Kalle et al. (2020). For example, we find that for the Komornik-Loreti constant $β\approx 1.78723$ we have $τ(β)=(2-β)/(β-1)$. Furthermore, we show that (i) the function $τ: β\mapstoτ(β)$ is left continuous on $(1,2]$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $τ$ has no downward jumps, with $τ(1+)=0$ and $τ(2)=1/2$; and (iii) there exists an open set $O\subset(1,2]$, whose complement $(1,2]\setminus O$ has zero Hausdorff dimension, such that $τ$ is real-analytic, convex and strictly decreasing on each connected component of $O$. Our strategy to find the critical value $τ(β)$ depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems. |
| title | Critical values for the $β$-transformation with a hole at $0$ |
| topic | Dynamical Systems Classical Analysis and ODEs Combinatorics Number Theory |
| url | https://arxiv.org/abs/2109.10012 |