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We study topological entropy of compactly supported Hamiltoniandiffeomorphisms from a perspective of persistence homology and Floer theory. Weintroduce barcode entropy, a Floer-theoretic invariant of a Hamiltoniandiffeomorphism, measuring exponential growth under iterations of the number ofnot-too-short bars in the barcode of the Floer complex. We prove that thebarcode entropy is bounded from above by the topological entropy and,conversely, that the barcode entropy is bounded from below by the topologicalentropy of any hyperbolic invariant set, e.g., a hyperbolic horseshoe. As aconsequence, we conclude that for Hamiltonian diffeomorphisms of surfaces thebarcode entropy is equal to the topological entropy.

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective