| Abstract: |
| The $\beta$-transformation $T_\beta(x):=\beta x\!\!\mod 1$, where $\beta>1$ is a real constant, has been studied by many authors due to its rich spectrum of dynamical properties. In 2018, Kalle, Kong, Langeveld and Li considered the $\beta$-transformation with $\beta\in(1,2]$ and a hole $(0,t)$, and studied the survivor set $K_\beta(t)$ of points whose forward orbit never enters the hole. Extending a well-known result of Urba\`nski (1987), they showed that for fixed $\beta$, the Hausdorff dimension of $K_\beta(t)$ is a decreasing devil`s staircase as a function of $t$, but left open the question of where this function reaches the value $0$. In recent work with D. Kong, we gave a complete answer to this question, which I will describe in this talk. I will also discuss a conjecture of Kalle {\em et al.} regarding the local dimension of the bifurcation set of the set-valued function $t\mapsto K_\beta(t)$ and, if time permits, explain a connection with topologically expansive Lorenz maps. (Joint work with D. Kong.) |
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