One-sided topological conjugacy of topological Markov shifts, continuous full groups and Cuntz–Krieger algebras

The paper’s Theorem 1.4 states the equivalence of (i) topological conjugacy, (ii) diagonal-preserving and potential-covariant isomorphisms of Cuntz–Krieger algebras, (iii) preservation of all cocycle fixed-point algebras, and (iv) a spatial isomorphism of continuous full groups that preserves all cocycle full subgroups; the proof chain (i)⇔(ii), (ii)⇒(iii), (iii)⇒(iv), (iv)⇒(i) is laid out explicitly (with (iv)⇒(i) proved in Section 4) and is internally consistent . The candidate solution proves the same equivalence via a groupoid route (Deaconu–Renault groupoids, cocycles, and Cartan reconstruction), invoking standard identifications and known rigidity (e.g., Brix–Carlsen). Its steps align with the paper’s statements and known references, but constitute a different proof strategy. I find no missing hypotheses relative to the paper’s assumptions (irreducible, non-permutation 0–1 matrices).