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Critical values for the $β$-transformation with a hole at $0$

Pieter Allaart, Derong Kong

Abstract

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.

Critical values for the $β$-transformation with a hole at $0$

Abstract

Given , let be the -transformation on the unit circle such that . For each let be the survivor set consisting of all whose orbit never hits the open interval . Kalle et al. proved in [Ergodic Theory Dynam. Systems, 40 (9): 2482--2514, 2020] that the Hausdorff dimension function is a non-increasing Devil's staircase. So there exists a critical value such that if and only if . In this paper we determine the critical value for all , answering a question of Kalle et al. (2020). For example, we find that for the Komornik-Loreti constant we have . Furthermore, we show that (i) the function is left continuous on with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) has no downward jumps, with and ; and (iii) there exists an open set , whose complement has zero Hausdorff dimension, such that is real-analytic, convex and strictly decreasing on each connected component of . Our strategy to find the critical value depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems.

Paper Structure

This paper contains 14 sections, 41 theorems, 209 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Farey words and Farey intervals
  3. Farey words
  4. Quasi-greedy expansions, Farey intervals and Lyndon intervals
  5. Greedy expansions and the symbolic survivor set
  6. Substitution of Lyndon words
  7. An equivalent definition of the substitution
  8. Properties of the substitution
  9. Critical values in a basic interval
  10. Geometrical structure of the basic intervals and exceptional sets
  11. Tree structure of the Lyndon intervals and relative exceptional sets
  12. The infinitely Farey set
  13. Critical values in the exceptional sets
  14. Càdlàg property of the critical value function

Key Result

Theorem 1

Figures (2)

  • Figure 1: The graph of the critical value function $\tau(\beta)$ for $\beta\in(1,2]$. We see that $\tau(\beta)\le 1-1/\beta$ for all $\beta\in(1,2]$, and the function $\tau$ is strictly decreasing in each basic interval $I^ {\mathbf{S}}$. For example, the basic interval generated by the Farey word $01$ is given by $I^{01}=[\beta_\ell, \beta_*]\approx[1.61803,1.73867]$ with $((10)^\infty)_{\beta_\ell}=(1100(10)^\infty)_{\beta_*}=1$. Furthermore, for any $\beta\in I^{01}$ we have $\tau(\beta)=(00(10)^\infty)_\beta=\frac{1}{\beta(\beta^2-1)}$; see Example \ref{['ex:cri-farey']} for more details.
  • Figure 2: The directed graph $G=(V, E)$ with the edge labels from $\left\{0, 1\right\}$ and vertex labels from $\left\{{ {\mathbf{s}}^-}, {\mathbf{s}}, \mathbf{a}, \mathbf{a}^+\right\}$, where ${\mathbf{s}}\in\Omega_{L}^*$ and $\mathbf{a}=\mathbb{L}( {\mathbf{s}})$.

Theorems & Definitions (82)

  • Theorem 1
  • Definition 1.1
  • Example 1.2
  • Proposition 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 2
  • Remark 1.6
  • Example 1.7
  • Definition 1.8
  • ...and 72 more