Phase transitions for unique codings of fat Sierpinski gaskets with multiple digits

Abstract

Given an integer $M\ge 1$ and $\beta\in(1, M+1)$, let $S_{\beta, M}$ be the fat Sierpinski gasket in $\mathbb R^2$ generated by the iterated function system $\left\{f_d(x)=\frac{x+d}{\beta}: d\in\Omega_M\right\}$, where $\Omega_M=\{(i,j)\in\mathbb Z_{\ge 0}^2: i+j\le M\}$. Then each $x\in S_{\beta, M}$ may be represented as a series $x=\sum_{i=1}^\infty\frac{d_i}{\beta^i}=:\Pi_\beta((d_i))$, and the infinite sequence $(d_i)\in\Omega_M^{\mathbb N}$ is called a \emph{coding} of $x$. Since $\beta \beta_c(M)$ then $U_{\beta, M}$ has positive Hausdorff dimension. Our results can also be applied to the intrinsic univoque set $\widetilde{U}_{\beta, M}$. Moreover, we show that the first critical base $\beta_G(M)$ is a Perron number, while the second critical base $\beta_c(M)$ is a transcendental number.