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- W3200804543 abstract "Given $betain(1,2]$, let $T_beta$ be the $beta$-transformation on the unit circle $[0,1)$ such that $T_beta(x)=beta xpmod 1$. For each $tin[0,1)$ let $K_beta(t)$ be the survivor set consisting of all $xin[0,1)$ whose orbit ${T^n_beta(x): nge 0}$ 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 $tmapstodim_H K_beta(t)$ is a non-increasing Devil's staircase. So there exists a critical value $tau(beta)$ such that $dim_H K_beta(t)>0$ if and only if $t<tau(beta)$. In this paper we determine the critical value $tau(beta)$ for all $betain(1,2]$, answering a question of Kalle et al. (2020). For example, we find that for the Komornik-Loreti constant $betaapprox 1.78723$ we have $tau(beta)=(2-beta)/(beta-1)$. Furthermore, we show that (i) the function $tau: betamapstotau(beta)$ is left continuous on $(1,2]$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $tau$ has no downward jumps, with $tau(1+)=0$ and $tau(2)=1/2$; and (iii) there exists an open set $Osubset(1,2]$, whose complement $(1,2]setminus O$ has zero Hausdorff dimension, such that $tau$ is real-analytic, convex and strictly decreasing on each connected component of $O$. Consequently, the dimension $dim_H K_beta(t)$ is not jointly continuous in $beta$ and $t$. Our strategy to find the critical value $tau(beta)$ depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems." @default.
- W3200804543 created "2021-09-27" @default.
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- W3200804543 date "2021-09-21" @default.
- W3200804543 modified "2023-09-27" @default.
- W3200804543 title "Critical values for the $beta$-transformation with a hole at $0$" @default.
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