SYMBOLIC DYNAMICS FOR NONUNIFORMLY HYPERBOLIC MAPS WITH SINGULARITIES IN HIGH DIMENSION

The paper’s Theorem 1.8 asserts a locally compact countable TMS and a Hölder coding π with π∘σ=f̂∘π, full measure of NUH# for all f–adapted χ–hyperbolic measures, and crucially that the coding restricted to the recurrent set is surjective and finite-to-one . The proof strategy explicitly builds a preliminary coding (Σ,σ,π) via ε–double charts and stable/unstable sets, notes this preliminary π can be infinite-to-one (hence does not yet satisfy Theorem 1.8), then constructs a locally finite cover Z and refines it to a Markov partition R to obtain a new coding (Σ̂,σ̂,π̂) that is finite-to-one and satisfies Theorem 1.8 . The candidate solution correctly reproduces most ingredients (Q(x̂), ε–double charts, q, qs, qu, shadowing, Hölder π, local compactness) and the measure-theoretic full-NUH# property , but it incorrectly claims finite-to-one directly for the coarse coding Σ using a “big return times” argument, omitting the paper’s essential refinement to a Markov partition that is needed to guarantee finite-to-one in this non-invertible/singular setting . Therefore, while much of the outline aligns with the paper, the key finite-to-one conclusion is asserted at the wrong stage and without the refinement that the paper proves.