A group from a map and orbit equivalence

The paper proves that any piecewise orientation-preserving circle map Φ satisfying (SE), (E±), (EC), (CS) gives rise to a discrete surface group G_Φ whose action is orbit equivalent to Φ, via a new construction of a hyperbolic dynamical space and a carefully defined group action, with an alternative direct topological proof using a 2-complex (see the main theorem and its strategy, as well as the orbit-equivalence proof in §6) . By contrast, the model’s solution assumes that one can glue a 2N-gon from the combinatorics and then directly obtain a Bowen–Series boundary map T_Γ with exactly the same (E±) and (EC) kneading data as Φ, from which a topological conjugacy h: S^1→S^1 is constructed. This core step contradicts the paper’s explicit remark that the original Bowen–Series map does not satisfy (EC) in general, hence one cannot simply assert that T_Γ has (EC) in the required way; the claimed Φ–T_Γ conjugacy and the subsequent identification G_Φ = h^{-1}Γh are therefore unsupported in general . The paper’s argument is complete and internally consistent; the model’s proof hinges on an unjustified equivalence to a Bowen–Series map and overlooks non-Markov identifications the paper handles via its dynamical graph and relations.