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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. 

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.