∆-revolving sequences and self-similar sets in the plane

The paper’s Theorem 3.3 states exactly the target identity X_{α,S} = ⋃_{γ∈Δ} γ·T_{α,S} for a generator set S with θ0=0 and |α|<1, where T_{α,S} is the attractor of the IFS ψ0(z)=αz+c0 and ψk(z)=(αe^{iθk})z+ck for k≥1. The proof in the paper partitions by the initial group element, defines X1,α,S with γ1 fixed, shows X1,α,S = T_{α,S} via a coding map arising from Lemma 3.1’s series representation, and then takes the union over γ∈Δ, concluding X_{α,S} = ⋃γ γ·T_{α,S} (Theorem 3.3 and its proof steps appear across these excerpts: definition of sΣ and X_{α,S}, statement of Theorem 3.3, the series representation of T_{α,S}, the definition of X1,α,S and the equivalence X1,α,S=T_{α,S}, and the final union over Δ) . The candidate solution follows the same structure: it partitions X_{α,S} by γ1, constructs the IFS coding set C and shows C is the self-similar attractor via invariance/contractivity, and identifies Y_γ with γ·T_{α,S} through the same coding correspondence, hence concluding X_{α,S} = ⋃γ γ·T_{α,S}. This mirrors the paper’s approach (the paper invokes Lemma 3.1 to derive the series form and then the coding, whereas the candidate directly verifies the IFS invariance and uniqueness), but the logical steps align one-to-one with the paper’s proof. The foundational definitions of Δ, W_Δ, and sΣ in the paper match the assumptions used by the model solution, including the rational-angle hypothesis ensuring Δ is finite and θ0=0 for the “stay” move . Overall, both are correct with substantially the same argument.