Homomorphisms from Aperiodic Subshifts to Subshifts with the Finite Extension Property

The uploaded paper proves that if every finitely generated subgroup of G has polynomial growth, X is aperiodic, and Y has the finite extension property (FEP), then a homomorphism X → Y exists (Theorem 1.4) . The proof proceeds by: (i) producing a disjoint-and-covering marker factor from X via a sliding-block code (Proposition 4.3) ; (ii) using the doubling property to bound overlap multiplicity (Lemma 4.4) ; (iii) defining nested, trimmed layers U_m and V_m culminating in V_{m0}=G (Lemma 5.3) ; and (iv) inductively filling patterns with a finite, translation-covariant patch library built from Y and applying FEP at each stage, yielding a sliding-block homomorphism (Section 5.2) . The candidate solution follows the same architecture: a marker factor with disjoint tiles whose inflations cover, a multiplicity bound from doubling, exactly-j overlap layers with a K-trim, a finite coherent patch library, and an FEP-driven induction to define a sliding-block code. Aside from a minor expositional gap in the base step (they suggest placing a single symbol without ensuring it is locally allowed), their construction matches the paper’s and can be fixed by selecting base patches as restrictions of a global point in Y, exactly as done in the paper’s Φ-construction with F=∅ . Therefore, both are correct and substantially the same proof.