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

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 $\{f_d(x)=\frac{x+d}{\beta }:d\in {\mathrm{\Omega }}_M\}$, where ${\mathrm{\Omega }}_M=\{(i,j)\in {\mathbb{Z}}_{\ge 0}^2:i+j\le M\}$. Then each $x\in S_{\beta ,M}$ may be represented as a series $x={\sum }_{i=1}^{\mathrm{\infty }}\frac{d_i}{{\beta }^i}=:{\mathrm{\Pi }}_{\beta }((d_i))$, and the infinite sequence $(d_i)\in {\mathrm{\Omega }}_M^{\mathbb{N}}$ is called a \emph{coding} of $x$. Since $\beta <M+1$, a point in $S_{\beta ,M}$ may have multiple codings. Let $U_{\beta ,M}$ be the set of $x\in S_{\beta ,M}$ having a unique coding, that is $$U_{\beta ,M}=\{x\in S_{\beta ,M}:\mathrm{\#}{\mathrm{\Pi }}_{\beta }^{-1}(x)=1\}\mathrm{.}$$ When $M=1$, Kong and Li [2020, Nonlinearity] described two critical bases for the phase transitions of the intrinsic univoque set ${\tilde{U}}_{\beta ,1}$, which is a subset of $U_{\beta ,1}$. In this paper we consider $M\ge 2$, and characterize the two critical bases ${\beta }_G(M)$ and ${\beta }_c(M)$ for the phase transitions of $U_{\beta ,M}$: (i) if $\beta \in (1,{\beta }_G(M)]$, then $U_{\beta ,M}$ is finite; (ii) if $\beta \in ({\beta }_G(M),{\beta }_c(M))$ then $U_{\beta ,M}$ is countably infinite; (iii) if $\beta ={\beta }_c(M)$ then $U_{\beta ,M}$ is uncountable and has zero Hausdorff dimension; (iv) if $\beta >{\beta }_c(M)$ then $U_{\beta ,M}$ has positive Hausdorff dimension. Our results can also be applied to the intrinsic univoque set ${\tilde{U}}_{\beta ,M}$. 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.