Phase transitions for unique codings of fat Sierpinski gaskets

Given an integer M ⩾ 1 and β ∈ (1, M + 1), let S_β,M be the fat Sierpinski gasket in ℝ² generated by the iterated function system {fd(x)=x+dβ:d∈ΩM}, where ΩM={(i,j)∈Z⩾02:i+j⩽M}. Then each x ∈ S_β,M may be represented as a series x=∑i=1∞diβi=:Πβ((di)), and the infinite sequence (d_i) ∈ Ω_M^ℕ is called a 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 ∈ S_β,M having a unique coding, i.e., Uβ,M={x∈Sβ,M:#Πβ−1(x)=1}. When M = 1, Kong and Li (2020) described two critical bases for the phase transitions of the intrinsic univoque set U~β,1, which is a subset of U_β,1. In this paper, we consider M ⩾ 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 ∈ (1, β_G(M)], then U_β,M is finite; (ii) if β ∈ (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. 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.