Operators of stochastic adding machines and Julia sets

The paper’s main dichotomy for the spectrum of S̃ on C(Γ_{d̄})—namely σ(S̃)=E_{d̄,p̄} if ∏p_r=0 and σ(S̃)=∂E_{d̄,p̄} if ∏p_r>0—and the equality σ=σ_ap are stated clearly in Theorem 4.1 and supported by the proof plan using a renormalization lemma (Lemma 5.3) and the transient/null-recurrent split (Proposition 5.6), which we find consistent and correct . However, Theorem 4.1 also states, “In particular … σ_pt(S̃)=⋃_r f̃_r^{-1}({1}), and ⋃_r f̃_r^{-1}({1})=∂E_{d̄,p̄}” (Eq. (4.3)), which contradicts both (i) the paper’s own summary that the eigenvalues form a countable dense subset of ∂E_{d̄,p̄} (hence only the closure equals ∂E_{d̄,p̄}), and (ii) standard cardinality considerations (e.g., for constant d and p, the boundary is an uncountable circle while ⋃_r f̃_r^{-1}({1}) is countable) . Earlier, Proposition 5.1 correctly proves ⋃_r f̃_r^{-1}({1})⊂σ_pt(S̃) and cl(⋃_r f̃_r^{-1}({1}))⊂∂E_{d̄,p̄}, which aligns with the density claim; the later equality (4.3) appears to be a misstatement/typo that cannot be literally true in general . The candidate (model) solution reproduces the main spectral dichotomy and corrects this point-spectrum statement to the closure equality cl(⋃_r f̃_r^{-1}({1}))=∂E_{d̄,p̄}, which is the only consistent formulation.