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 locally maximal 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.
In the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic orbits. These maps are of interest in both dynamics and symplectic topology. We show that a closed, monotone symplectic manifold, which admits a nondegenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element and hence nontrivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on nonvanishing Gromov–Witten invariants of manifolds admitting pseudo-rotations.
In the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic orbits. These maps are of interest in both dynamics and symplectic topology. We show that a closed, monotone symplectic manifold, which admits a nondegenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element, and hence non-trivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on non-vanishing Gromov-Witten invariants of manifolds admitting pseudo-rotations.
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