Coded equivalence of one-sided topological Markov shifts

The paper’s main theorem states that coded equivalence implies continuous orbit equivalence (COE) for one-sided topological Markov shifts, and proves it by (i) showing that for any right Markov code C, the standard coding homeomorphism h_C implements a COE between (X_A,σ_A) and (X_{A(C)},σ_{A(C)}) (Proposition 13), and (ii) noting COE is an equivalence relation; thus any elementary coded equivalence and finite chains yield COE (Theorem 14). The candidate solution reproduces this structure: Step 1 constructs h_C via unique factorization; Steps 2–3 build continuous cocycles from the shift-invariance and word-length data of C; Step 5 passes from elementary coded equivalence to COE; Step 6 composes along a chain, with an explicit cocycle-composition formula. These steps match the paper’s Proposition 13 construction and the Theorem 14 argument, differing only in that the candidate writes out a composition formula while the paper cites the equivalence property from prior work. Therefore both are correct and essentially the same proof. See the definitions of right Markov code and the standard coding homeomorphism, the COE construction in Proposition 13, and Theorem 14’s conclusion that coded equivalence implies COE .