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

Given $\beta \in (1,2]$, let $T_{\beta }:[0,1)\to [0,1);x\mapsto \beta x(\mathrm{m}\mathrm{o}\mathrm{d}1)$. For $m\in \mathbb{N}$ let $${\tau }_m(\beta ):=\sup \{t\in [0,1):K_{\beta }(t)\text{ contains a periodic orbit of smallest period }m\},$$ where $K_{\beta }(t)=\{x\in [0,1):T_{\beta }^n(x)\mathrm{\notin }(0,t)\mathrm{\forall }n\ge 0\}$ is the survivor set of the open dynamical system $(T_{\beta },[0,1),H)$ with a hole $H=(0,t)$. In this paper we give a complete characterization of ${\tau }_m$, and show that ${\tau }_m$ is piecewise continuous with precisely $\psi (m)$ discontinuity points, where $\psi (m)$ is the number of bulbs of period $m$ in the Mandelbrot set. To describe the critical value function ${\tau }_m$ we construct a finite butterfly tree ${\mathcal{T}}_m$, from which we are able to determine the discontinuity points and the analytic formula of ${\tau }_m$ based on Farey words and substitution operators. As a by product, we characterize the extremal Lyndon words and extremal Perron words. Since we are working in the symbolic space, our result can be applied to study phase transitions for periodic orbits in topologically expansive Lorenz maps, doubling map with an asymmetric hole, intermediate $\beta$-transformations, unique expansions in double bases, and so on.