Periodic unique codings of fat Sierpinski gasket

For $\beta >1$ let $S_{\beta }$ be the Sierpinski gasket generated by the iterated function system $$\{f_{{\alpha }_0}(x,y)=(\frac{x}{\beta },\frac{y}{\beta }),f_{{\alpha }_1}(x,y)=(\frac{x+1}{\beta },\frac{y}{\beta }),f_{{\alpha }_2}(x,y)=(\frac{x}{\beta },\frac{y+1}{\beta }\left)\right\}\mathrm{.}$$ If $\beta \in (1,2]$, then the overlap region $O_{\beta }:={\bigcup }_{i\mathrm{\ne }j}{f}_{{\alpha }_i}({\mathrm{\Delta }}_{\beta })\cap f_{{\alpha }_j}({\mathrm{\Delta }}_{\beta })$ is nonempty, where ${\mathrm{\Delta }}_{\beta }$ is the convex hull of $S_{\beta }$. In this paper we study the periodic codings of the univoque set $${\mathbf{U}}_{\beta }:=\{(d_i{)}_{i=1}^{\mathrm{\infty }}\in \{(0,0),(1,0),(0,1){\}}^{\mathbb{N}}:\sum _{i=1}^{\mathrm{\infty }}{d}_{n+i}{\beta }^{-i}\in S_{\beta }\setminus O_{\beta }\mathrm{\forall }n\ge 0\}\mathrm{.}$$ 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 ${\beta }_{3k}>{\beta }_{3\mathrm{\ell }}$ if and only if $k$ is larger than $\mathrm{\ell }$ in the Sharkovskii ordering; and the sequences $({\beta }_{3\mathrm{\ell }+1}),({\beta }_{3\mathrm{\ell }+2})$ 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 $k\in \mathbb{N}$.