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

Abstract: Given $\beta\in(1,2]$, let $T_\beta$ be the $\beta$-transformation on the unit circle $[0,1)$ such that $T_\beta(x)=\beta x\pmod 1$. For each $t\in[0,1)$ let $K_\beta(t)$ be the survivor set consisting of all $x\in[0,1)$ whose orbit ${T^n_\beta(x): n\ge 0}$ never enters the interval $[0,t)$. Letting $\mathscr{E}\beta$ 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}\beta\cap[t,1]\big)=\dim_H K\beta(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 $\mathscr{E}\beta$ with the bifurcation set $\mathscr{B}\beta$ 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 $\beta\in(1,2]$, the difference $\mathscr{E}\beta\backslash\mathscr{B}\beta$ 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 $\mathscr{E}\beta\backslash\mathscr{B}\beta$ 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.