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In this paper we continue investigating connections between Floer theory and dynamics of Hamiltonian systems, focusing on the barcode entropy of Reeb flows. Barcode entropy is the exponential growth rate of the number of not-too-short bars in the Floer or symplectic homology persistence module. The key novel result is that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set. This, combined with the fact that the topological entropy bounds the barcode entropy from above, established by Fender, Lee and Sohn, implies that in dimension three the two types of entropy agree. The main new ingredient of the proof is a variant of the Crossing Energy Theorem for Reeb flows. 

We study lower semi-continuity properties of the volume, i.e., the surface area, of a closed Lagrangian manifold with respect to the Hofer- and γ-distance on a class of monotone Lagrangian submanifolds Hamiltonian isotopic to each other. We prove that volume is γ-lower semi-continuous in two cases. In the first one the volume form comes from a Kähler metric with a large group of Hamiltonian isometries, but there are no additional constraints on the Lagrangian submanifold. The second one is when the volume is taken with respect to any compatible metric, but the Lagrangian submanifold must be a torus. As a consequence, in both cases, the volume is Hofer lower semi-continuous. 

We introduce and study the barcode entropy for geodesic flows of closed Riemannian manifolds, which measures the exponential growth rate of the number of not-too-short bars in the Morse-theoretic barcode of the energy functional. We prove that the barcode entropy bounds from below the topological entropy of the geodesic flow and, conversely, bounds from above the topological entropy of any hyperbolic compact invariant set. As a consequence, for Riemannian metrics on surfaces, the barcode entropy is equal to the topological entropy. A key to the proofs and of independent interest is a crossing energy theorem for gradient flow lines of the energy functional. 

We study topological entropy of compactly supported Hamiltonian diffeomorphisms from a perspective of persistent 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. 

Our main result is that for any closed symplectic manifold, the spectral norm of the iterates of a Hamiltonian diffeomorphism is locally uniformly bounded away from zero C-infinity generically. 

This paper is a follow-up to the authors’ recent work on barcode entropy. We study the growth of the barcode of the Floer complex for the iterates of a compactly supported Hamiltonian diffeomorphism. In partic- ular, we introduce sequential barcode entropy which has properties similar to barcode entropy, bounds it from above and is more sensitive to the bar- code growth. In the same vein, we explore another variant of barcode entropy based on the total persistence growth and revisit the relation between the growth of periodic orbits and topological entropy. We also study the behav- ior of the spectral norm, aka the γ-norm, under iterations. We show that the γ-norm of the iterates is separated from zero when the map has sufficiently many hyperbolic periodic points and, as a consequence, it is separated from zero C ∞-generically in dimension two. We also touch upon properties of the barcode entropy of pseudo-rotations and, more generally, γ-almost periodic maps.