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

TL;DR

This paper investigates phase transitions in the set of points with unique codings in fat Sierpinski gaskets generated by multiple digits, identifying critical bases where the structure of this set changes dramatically.

Contribution

It characterizes two critical bases for phase transitions in the univoque set of fat Sierpinski gaskets with multiple digits, extending prior work from the case M=1 to M≥2.

Findings

The set U_{β,M} is finite for β ≤ β_G(M).

U_{β,M} becomes uncountably infinite with zero Hausdorff dimension at β=β_c(M).

β_G(M) is a Perron number, β_c(M) is transcendental.

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.