Phase transitions on periodic orbits in $\beta$-transformation with a hole at zero

Derong Kong, Dantong Pu

Given $\beta \in (1,2]$, let $T_{\beta }:[0,1)\to [0,1);x\mapsto \beta x(\mathrm{mod}1)$. For $m\in \mathbb{N}$ let \[ τ_m(β):=\sup\left\{t\in[0,1): K_β(t)\textrm{ {contains a periodic orbit} of smallest peri…