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

Given an integer $M\ge 1$ and $β\in(1, M+1)$, let $S_{β, M}$ be the fat Sierpinski gasket in $\mathbb R^2$ generated by the iterated function system $\left\{f_d(x)=\frac{x+d}β: d\inΩ_M\right\}$, where $Ω_M=\{(i,j)\in\mathbb Z_{\ge 0}^2: i+j\le M\}$. Then each $x\in S_{β, M}$ may be represented as a series $x=\sum_{i=1}^\infty\frac{d_i}{β^i}=:Π_β((d_i))$, and the infinite sequence $(d_i)\inΩ_M^{\mathbb N}$ is called a \emph{coding} of $x$. Since $β<M+1$, a point in $S_{β, M}$ may have multiple codings. Let $U_{β, M}$ be the set of $x\in S_{β, M}$ having a unique coding, that is $U_{β, M}=\left\{x\in S_{β, M}: \#Π_β^{-1}(x)=1\right\}.$ When $M=1$, Kong and Li [2020, Nonlinearity] described two critical bases for the phase transitions of the intrinsic univoque set $\widetilde U_{β, 1}$, which is a subset of $U_{β, 1}$. In this paper we consider $M\ge 2$, and characterize the two critical bases $β_G(M)$ and $β_c(M)$ for the phase transitions of $U_{β, M}$: (i) if $β\in(1, β_G(M)]$, then $U_{β, M}$ is finite; (ii) if $β\in(β_G(M), β_c(M))$ then $U_{β, M}$ is countably infinite; (iii) if $β=β_c(M)$ then $U_{β, M}$ is uncountable and has zero Hausdorff dimension; (iv) if $β>β_c(M)$ then $U_{β, M}$ has positive Hausdorff dimension. Our results can also be applied to the intrinsic univoque set $\widetilde{U}_{β, M}$. Moreover, we show that the first critical base $β_G(M)$ is a Perron number, while the second critical base $β_c(M)$ is a transcendental number.

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