  We study topological entropy of compactly supported Hamiltonian
diffeomorphisms from a perspective of persistence homology and Floer theory. We
introduce barcode entropy, a Floer-theoretic invariant of a Hamiltonian
diffeomorphism, measuring exponential growth under iterations of the number of
not-too-short bars in the barcode of the Floer complex. We prove that the
barcode entropy is bounded from above by the topological entropy and,
conversely, that the barcode entropy is bounded from below by the topological
entropy of any hyperbolic invariant set, e.g., a hyperbolic horseshoe. As a
consequence, we conclude that for Hamiltonian diffeomorphisms of surfaces the
barcode entropy is equal to the topological entropy.
