Entropy plateaus, transitivity and bifurcation sets for the $\beta$-transformation with a hole at $0$

Abstract

Given $\beta >1$, let $T_{\beta }$ be the $\beta$-transformation on the unit circle $[0,1)$ such that $T_{\beta }(x)=\beta x(\mathrm{mod}1)$. For each $t\in [0,1)$ let $K_{\beta }(t)$ be the survivor set consisting of all $x\in [0,1)$ whose orbit $\{T_{\beta }^n(x):n\ge 0\}$ never enters the interval $[0,t)$. Letting ${\mathcal{E}}_{\beta }$ denote the bifurcation set of the set-valued map $t\mapsto K_{\beta }(t)$, Kalle et al. [Ergodic Theory Dynam. Systems, 40 (9): 2482--2514, 2020] conjectured that \[ \dim_H\big(\mathscr{E}_β\cap[t,1]\big)=\dim_H K_β(t) \qquad \forall\,t\in(0,1). \] The main purpose of this article is to prove this conjecture. We do so by investigating dynamical properties of the symbolic equivalent of the survivor set $K_{\beta }(t)$, in particular its entropy and topological transitivity. In addition, we compare ${\mathcal{E}}_{\beta }$ with the bifurcation set ${\mathcal{B}}_{\beta }$ of the map $t\mapsto {\dim }_H{K}_{\beta }(t)$ (which is a decreasing devil's staircase by a theorem of Kalle et al.), and show that, for Lebesgue-almost every $\beta >1$, the difference ${\mathcal{E}}_{\beta }\backslash {\mathcal{B}}_{\beta }$ has positive Hausdorff dimension, but for every $k\in \{0,1,2,\dots \}\cup \{{\aleph }_0\}$, there are infinitely many values of $\beta$ such that the cardinality of ${\mathcal{E}}_{\beta }\backslash {\mathcal{B}}_{\beta }$ is exactly $k$. For a countable but dense subset of $\beta$'s, we also determine the intervals of constancy of the function $t\mapsto {\dim }_H{K}_{\beta }(t)$. Some connections with other topics in dynamics, such as kneading invariants of Lorenz maps and the doubling map with an arbitrary hole, are also discussed.

arXiv comment73 pages, 1 figure. We have extended the results to all beta>1