Periodic points in the $β$-transformation with a hole at 0
Yuzheng Bi
TL;DR
The paper investigates periodic points of the beta-transformation $T_\beta(x)=\{\beta x\}$ on $[0,1)$ with a hole at the origin, focusing on the survivor set $K_\beta(t)$ and the maximal hole size $S_\beta(p)$ that still admits a point of smallest period $p$. It provides explicit formulas for $S_\beta(p)$ for three benchmark bases: $\beta=2$, $\beta=\phi_2$ (golden ratio), and $\beta=\phi_3$ (tribonacci), expressed in terms of specific infinite beta-expansions and their lexicographic order properties. The results include closed forms such as $S_2(p)=[(01^{p-1})^\infty]_2 = (2^{p-1}-1)/(2^p-1)$, and analogous families for the golden and tribonacci cases, with asymptotic limits $S_2(p)\to 1/2$, $S_{\phi_2}(p)\to 1/(\beta^3-\beta)$, and $S_{\phi_3}(p)\to (\beta^{2}+1)/(\beta^{4}-\beta)$ as $p\to\infty$. The methods rely on a detailed combinatorial analysis of survivor sets via greedy and quasi-greedy $\beta$-expansions, lexicographic order, and the structure of multinacci polynomials, yielding precise threshold sequences for each case. These results illuminate the bifurcation structure of surviving periodic points in non-integer base dynamical systems and quantify how hole size governs the appearance of periodic behavior across different $\beta$.
Abstract
For $β\in(1,2]$ let $T_β: [0,1)\to[0,1); x\mapsto βx\pmod 1$. In this paper we study the periodic points in the open dynamical system $([0,1), T_β)$ with a hole $[0,t)$. For $p\in\mathbb{N}$ we characterize the largest $t$, denoted by $S_β(p)$, in which the survivor set $K_β(t)$ has a periodic point of smallest period $p$. More precisely, we give precise formulae for this critical value $S_β(p)$ when $β=2$, $β=\frac{1+\sqrt{5}}{2}$ and $β$ being the tribonacci number. We show that for $β=2$ the critical value $S_2(p)$ converges to $1/2$ as $p\to \infty$. When $β=\frac{1+\sqrt{5}}{2}$, the critical value $S_β(p)\to \frac{1}{β^3-β}$. While $β$ is the tribinacci number, the critical value $S_β(p)\to \frac{β^{2}+1}{β^4-β}$.