Topological Entropy of Hamiltonian Diffeomorphisms: A Persistence Homology and Floer Theory Perspective

The paper proves three key statements: (i) an upper bound of barcode entropy by topological entropy (Theorem A and Corollary A), (ii) a lower bound on barcode entropy by the entropy of any locally maximal uniformly hyperbolic invariant set (Theorem B), and (iii) equality on closed surfaces by combining (i)+(ii) with Katok’s horseshoe approximation (Theorem C). These results are stated and sketched explicitly in the paper, including the absolute/relative definitions and the necessity of using all free homotopy classes in the absolute Floer complex (Remark 2.9 and Example 2.10) . The candidate solution reproduces this structure: Step 1 cites the upper bound; Step 2 cites the hyperbolic-set lower bound; Step 3 invokes Katok’s approximation on surfaces to conclude equality. Minor imprecisions are: (a) The proof of the upper bound in the paper hinges on a Lagrangian ‘tomograph’ plus Crofton-type counting and Yomdin’s theorem, rather than directly on an action–energy identity (though energy–action relations are used crucially in the lower bound) ; and (b) Theorem B assumes the hyperbolic set is locally maximal, which the candidate did not explicitly state. These do not affect the conclusion. Overall, the model’s argument matches the paper’s proof strategy and conclusions on surfaces.