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Variation on the Theme of the Conley Conjecture


We prove a generalization of the Conley conjecture: Every Hamiltonian diffeomorphism of a closed symplectic manifold has infinitely many periodic orbits if the first Chern class vanishes on the second fundamental group. In particular, this removes the rationality condition from similar theorems by Ginzburg and Gürel. The proof in the irrational case involves several new ingredients including the definition and the properties of the filtered Floer homology for Hamiltonians on irrational manifolds. For this proof, we develop a method of localizing the filtered Floer homology for short action intervals using a direct sum decomposition. One of the summands only depends on the behavior of the Hamiltonian in a fixed open set and enables us to use tools from more restrictive cases in the proof of the Conley conjecture. We also prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians, which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits.

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