Periodic unique codings of fat Sierpinski gasket

Abstract

For $\beta>1$ let $S_\beta$ be the Sierpinski gasket generated by the iterated function system \[\left\{f_{\alpha_0}(x,y)=\Big(\frac{x}{\beta},\frac{y}{\beta}\Big), \quad f_{\alpha_1}(x,y)=\Big(\frac{x+1}{\beta}, \frac{y}{\beta}\Big), \quad f_{\alpha_2}(x,y)=\Big(\frac{x}{\beta}, \frac{y+1}{\beta}\Big)\right\}.\] If $\beta\in(1,2]$, then the overlap region $O_\beta:=\bigcup_{i\ne j}f_{\alpha_i}(\Delta_\beta)\cap f_{\alpha_j}(\Delta_\beta)$ is nonempty, where $\Delta_\beta$ is the convex hull of $S_\beta$. In this paper we study the periodic codings of the univoque set \[ \mathbf U_\beta:=\left\{(d_i)_{i=1}^\infty\in\{(0,0), (1,0), (0,1)\}^\mathbb N: \sum_{i=1}^\infty d_{n+i}\beta^{-i}\in S_\beta\setminus O_\beta~\forall n\ge 0\right\}. \] More precisely, we determine for each $k\in\mathbb N$ the smallest base $\beta_k\in(1,2]$ such that for any $\beta>\beta_k$ the set $\mathbf U_\beta$ contains a sequence of smallest period $k$. We show that each $\beta_k$ is a Perron number, and the sequence $(\beta_k)$ has infinitely many accumulation points. Furthermore, we show that $\beta_{3k}>\beta_{3\ell}$ if and only if $k$ is larger than $\ell$ in the Sharkovskii ordering; and the sequences $ (\beta_{3\ell+1}), (\beta_{3\ell+2})$ decreasingly converge to the same limit point $\beta_a\approx 1.55898$, respectively. In particular, we find that $\beta_{6m+4}=\beta_{3m+2}$ for all $m\ge 0$. Consequently, we prove that if $\mathbf U_\beta$ contains a sequence of smallest period $2$ or $4$, then $\mathbf U_\beta$ contains a sequence of smallest period $k$ for any $k\in\mathbb N$.