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 $\beta<M+1$, 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 \[ U_{\beta, M}=\left\{x\in S_{\beta, M}: \#\Pi_\beta^{-1}(x)=1\right\}. \] When $M=1$, Kong and Li [2020, Nonlinearity] described two critical bases for the phase transitions of the intrinsic univoque set $\widetilde U_{\beta, 1}$, 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 $\widetilde{U}_{\beta, M}$. 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.