Periodic unique codings of fat Sierpinski gasket

Abstract

For $\beta >1$ let $S_{\beta }$ be the Sierpinski gasket generated by the iterated function system \[\left\{f_{α_0}(x,y)=\Big(\frac{x}β,\frac{y}β\Big), \quad f_{α_1}(x,y)=\Big(\frac{x+1}β, \frac{y}β\Big), \quad f_{α_2}(x,y)=\Big(\frac{x}β, \frac{y+1}β\Big)\right\}.\] 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 \[ \mathbf U_β:=\left\{(d_i)_{i=1}^\infty\in\{(0,0), (1,0), (0,1)\}^\mathbb N: \sum_{i=1}^\infty d_{n+i}β^{-i}\in S_β\setminus O_β~\forall n\ge 0\right\}. \] 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 $k\in \mathbb{N}$.

34 pages, 6 figures