Volume growth and topological entropy of certain partially hyperbolic systems

The uploaded paper’s Theorem A states that for C1 diffeomorphisms admitting a partially hyperbolic splitting with one‑dimensional center blocks, the topological entropy equals both the lim inf and the lim sup of (1/n) log ∫ maxV |det(Dfnx|V)| dx; thus the limit exists and equals h_top(f) (see the abstract and Theorem A) . The proof proceeds by (i) an upper bound h_top(f) ≤ lim inf 1/n log ∫ maxV … via Proposition 2.1 and Lemma 2.2 together with the variational principle and Katok’s spanning‑set characterization , and (ii) a lower bound h_top(f) ≥ lim sup 1/n log ∫ maxi |det(Dfnx|Fi)| dx via Lemma 2.7, Corollary 2.8, and Proposition 2.6, then bridging max over Fi to max over all subspaces with Proposition 3.1 (supported by Proposition 3.2) to conclude equality and existence of the limit . By contrast, the candidate solution mis-states the upper bound as a lim sup (it is a lim inf in Proposition 2.1) and, more critically, asserts a two‑sided, n‑uniform bounded distortion for integrals of An over Bowen balls (Step 2), citing Lemma 2.7. Lemma 2.7 provides only an upper bound, not a two‑sided comparability, and the paper explicitly builds the lower‑bound direction by a different mechanism (Corollary 2.4/2.8 and Proposition 2.6), avoiding any uniform lower bound on Bowen balls . Hence the paper’s argument is coherent and correct as written, while the candidate’s proof relies on unsupported claims and incorrect citations.