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

TL;DR

This work analyzes phase transitions of periodic orbits for the open β-transformation with a hole at zero by introducing the critical value function $τ_m(β)$. It builds a finite butterfly tree $\mathcal{T}_m$ and leverages Farey, Lyndon, and Perron word structures together with the substitution $\bullet$ to achieve a complete, piecewise-analytic description of $τ_m$, with exactly $ψ(m)$ discontinuities corresponding to Mandelbrot bulbs. The endpoints of the partition intervals are described via bases $β_{m,k_1,\dots,k_j}$ and explicit formulas express $τ_m(β)$ as β-expansions of infinite words $((\mathbf{w}_{k_1/m_1}\bullet\cdots\bullet\mathbf{w}_{k_j/m_j})^\infty)_β$, enabling an effective algorithm to compute $τ_m(β)$ for any $(m,β)$. The framework also connects to kneading theory for Lorenz maps and to other open dynamical systems, illustrating broad applicability to intermediate β-transformations and unique expansions in double bases. Overall, the paper provides a unified symbolic mechanism for phase transitions in open dynamical systems, linking combinatorics on words, number theory, and complex dynamics.

Abstract

Given $β\in(1,2]$, let $T_β: [0,1)\to[0,1);~x\mapstoβx\pmod 1$. For $m\in\mathbb N$ let \[ τ_m(β):=\sup\left\{t\in[0,1): K_β(t)\textrm{ {contains a periodic orbit} of smallest period }m \right\}, \] where $K_β(t)=\{x\in[0,1): T_β^n(x)\notin(0,t)~\forall n\ge 0\}$ is the survivor set of the open dynamical system $(T_β, [0,1), H)$ with a hole $H=(0,t)$. In this paper we give a complete characterization of $τ_m$, and show that $τ_m$ is piecewise continuous with precisely $ψ(m)$ discontinuity points, where $ψ(m)$ is the number of bulbs of period $m$ in the Mandelbrot set. To describe the critical value function $τ_m$ we construct a finite butterfly tree $\mathcal T_m$, from which we are able to determine the discontinuity points and the analytic formula of $τ_m$ 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 $β$-transformations, unique expansions in double bases, and so on.

Figures (6)

• Figure 1: The butterfly tree $\mathcal{T}_8$ has $9$ leaves. Each directed path from the root to a leaf corresponds to an admissible $8$-chain in $\mathcal{A}_8$.
• Figure 2: The graph of the arithmetic function $\psi(n)$ for $1\le n\le 100$.
• Figure 3: The butterfly tree $\mathcal{T}_8$ with Farey words as its vertices.
• Figure 4: The graph of the critical value function $\tau_8$, which has $\psi(8)=9$ discontinuity points.
• Figure 5: The directed graph $G_ {\mathbf{s}}=(V_ {\mathbf{s}}, E_ {\mathbf{s}})$.

Theorems & Definitions (71)

• Example 1.1
• Proposition 1.2
• proof
• Theorem 1.3
• Remark 1.4
• Example 1.5
• Theorem 1.6
• Remark 1.7
• Lemma 2.1
• Lemma 2.2