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

TL;DR

This work develops a complete phase-transition framework for unique codings on fat Sierpinski gaskets $S_{β,M}$ with $M\ge 2$ and $β\in(1,M+1)$. It introduces two critical bases, $β_G(M)$ and $β_c(M)$, and shows four regimes for the univoque sets $U_{β,M}$ and the intrinsic sets $\widetilde{U}_{β,M}$, with finiteness, countable infinitude, zero-dimension uncountability, and positive-dimension behavior as $β$ crosses these thresholds. The authors derive explicit formulas for $β_G(M)$ and $β_c(M)$ across three residue classes $M=3N+1,3N+2,3N+3$, with $β_G(M)$ always a Perron number and $β_c(M)$ always transcendental; the proofs rely on Thue–Morse–type constructions and two- and three-dimensional symbolic dynamics, including detailed block substitutions. They also establish phase transitions by constructing admissible subshifts of finite type, yielding positive Hausdorff dimension for $β>β_c(M)$ and zero dimension at $β=β_c(M)$, with explicit dimensions in multinacci cases. Overall, the results generalize the $M=1$ case and provide a unified combinatorial and number-theoretic framework for the intrinsic and standard univoque sets on fat Sierpinski gaskets.

Abstract

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.

Figures (7)

• Figure 1: The graph for the first generation of $S_{\beta, M}$ with $M=4$ and $\beta=4$. The convex hull $\Delta_{\beta, M}$ is the triangle with vertices $(0, 0), (4/(\beta-1), 0)$ and $(0, 4/(\beta-1))$. Each light grey triangle corresponds to a $f_{\beta, \alpha_{ij}}(\Delta_{\beta,M})$ for some $\alpha_{ij}\in {\Omega}_M$, and the overlap region $O_{\beta, M}$ is the union of eighteen small dark grey triangles.
• Figure 2: The graph of the two critical bases $\beta_G(M)$ and $\beta_c(M)$ with $M=1,2,\ldots, 10$.
• Figure 3: The typical pattern of $f_{\beta, d}(\Delta_{\beta,M})\setminus O_{\beta,M}$ with $d=(d^1, d^2)\in\Omega_M$ satisfying $d^1<M, d^2<M$ and $\overline{d^ {\oplus}}=M-(d^1+d^2)<M$.
• Figure 4: The three patterns of $f_{\beta, d}(\Delta_{\beta,M})\setminus O_{\beta,M}$ with $d=(0,0), d=(M,0)$ and $d=(0,M)$ respectively.
• Figure 5: The labeled graph $G$ presenting $(X_k, \sigma)$.

Theorems & Definitions (59)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: Sidorov_2007Kong-Li-2020
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Proposition 2.5