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

TL;DR

This paper characterizes the phase transition points for periodic orbits in beta-transformations with a hole at zero, revealing a piecewise structure linked to the Mandelbrot set and constructing a finite butterfly tree for analysis.

Contribution

It provides a complete characterization of the critical value function for periodic orbits in beta-transformations with a hole, including a finite butterfly tree construction and explicit formulas.

Findings

The function $ au_m$ is piecewise continuous with $ ext{psi}(m)$ discontinuities.

Discontinuity points are determined by a finite butterfly tree structure.

The results connect phase transitions in dynamical systems to Mandelbrot set properties.

Abstract

Given $\beta\in(1,2]$, let $T_\beta: [0,1)\to[0,1);~x\mapsto\beta x\pmod 1$. For $m\in\mathbb N$ let \[ \tau_m(\beta):=\sup\left\{t\in[0,1): K_\beta(t)\textrm{ {contains a periodic orbit} of smallest period }m \right\}, \] where $K_\beta(t)=\{x\in[0,1): T_\beta^n(x)\notin(0,t)~\forall n\ge 0\}$ 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.