MEASURES MAXIMIZING THE ENTROPY FOR KAN ENDOMORPHISMS

The paper proves Theorem A for Kan-like maps: h_top(K)=h_top(E), the MMEs are exactly the lifts of ν0, and—under (6)—there are exactly three ergodic MMEs (μ0, μ+, μ1) with the stated properties, including intermingled basins (w.r.t. ν0×LebI) and periodic-orbit approximation of μ+; the argument is complete and coherent. The candidate solution mirrors the paper’s structure: it uses Ledrappier–Walters to identify MMEs as lifts of ν0, constructs μ0, μ1 with negative center exponents, establishes intermingled basins and full measure of B(μ0)∪B(μ1), and cites the same result for existence/uniqueness of interior MME μ+ with λc>0. The only substantive divergence is Step 6, where the candidate invokes Katok’s periodic-orbit approximation via the natural extension; while the conclusion matches the paper, that justification is not obviously applicable (the natural extension is a hyperbolic homeomorphism/cocycle, not a C^{1+} diffeomorphism on a manifold). Replacing that step with the paper’s Lemmas 4.2–4.3 resolves the gap. Overall, both arrive at the same results with essentially the same backbone proof; the model’s Step 6 needs a citation adjustment, not a change in conclusion.