The $β$-transformation with a hole at $0$: the general case

TL;DR

This paper extends the theory of β-transformations with holes from the well-studied range $\beta\in(1,2]$ to all $\beta>1$, focusing on survivor sets $K_\beta(t)$ and the bifurcation set $\mathscr{E}_\beta$. It introduces extended Farey words and a substitution operator to manage larger alphabets, enabling explicit descriptions of the critical value $\tau(\beta)$ across basic intervals, exceptional sets, and infinite renormalization regimes, with detailed regularity properties including left-continuity with countably many jumps and analyticity on open components. The authors prove that $\mathscr{E}_\beta$ is Lebesgue-null yet of full Hausdorff dimension, and show rich fine structure in $\mathscr{E}_\beta$ for almost every $\beta$, including infinitely many isolated and accumulation points near zero, while identifying a zero-dimensional exceptional set $E_L$ where no isolated points occur. Finally, they reveal a deep connection between the survivor-set problem and the times-$k$ map with multiple holes, establishing a dimension-preserving correspondence between $K(a,b;k-1)$ and the symbolic survivor set, thus linking open dynamical systems with combinatorial constructions on extended Farey words.

Abstract

Given $β>1$, let $T_β$ be the $β$-transformation on the unit circle $[0,1)$, defined by $T_β(x)=βx-\lfloor βx\rfloor$. 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 interval $[0,t)$. Kalle et al.~[{\em Ergodic Theory Dynam. Systems} {\bf 40} (2020), no.~9, 2482--2514] considered the case $β\in(1,2]$. They studied the set-valued bifurcation set $\mathscr{E}_β:=\{t\in[0,1): K_β(t')\ne K_β(t)~\forall t'>t\}$ and proved that the Hausdorff dimension function $t\mapsto\dim_H K_β(t)$ is a non-increasing Devil's staircase. In a previous paper [{\em Ergodic Theory Dynam. Systems} {\bf 43} (2023), no.~6, 1785--1828] we determined, for all $β\in(1,2]$, the critical value $τ(β):=\min\{t>0: η_β(t)=0\}$. The purpose of the present article is to extend these results to all $β>1$. In addition to calculating $τ(β)$, we show that (i) the function $τ: β\mapstoτ(β)$ is left continuous on $(1,\infty)$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $τ$ has no downward jumps; and (iii) there exists an open set $O\subset(1,\infty)$, whose complement $(1,\infty)\backslash O$ has zero Hausdorff dimension, such that $τ$ is real-analytic, strictly convex and strictly decreasing on each connected component of $O$. We also prove several topological properties of the bifurcation set $\mathscr{E}_β$. The key to extending the results from $β\in(1,2]$ to all $β>1$ is an appropriate generalization of the Farey words that are used to parametrize the connected components of the set $O$. Some of the original proofs from the above-mentioned papers are simplified.

Figures (3)

• Figure 1: The graph of the critical value function $\tau(\beta)$ for $\beta\in(1,4]$, with some of the basic intervals marked by blue dotted lines. (See Section \ref{['sec:critical-value']} for the definitions.) The interiors of the basic intervals make up the set $O$ in Theorem \ref{['main:critical-devils-staircase']}. Black dotted lines indicate jumps in the graph.
• Figure 2: The directed graph illustrating the map $\Phi_ {\mathbf{s}}$.
• Figure 3: The open dynamical system $(T_3, [0,1), H)$ with $H=H_1\cup H_2$. Note that $H_2=H_1+\frac{1}{3}$.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Definition 2.2
• Example 2.3
• Remark 2.4
• Lemma 2.5
• Remark 2.6
• Proposition 3.1
• Theorem 3.2