# The Conley conjecture for irrational symplectic manifolds

## TL;DR

This paper extends the Conley conjecture to irrational symplectic manifolds, proving that Hamiltonian diffeomorphisms have infinitely many periodic orbits without requiring rationality conditions, using new Floer homology techniques.

## Contribution

It generalizes the Conley conjecture to irrational manifolds by developing filtered Floer homology methods applicable in this setting.

## Key findings

- Proves infinite periodic orbits for Hamiltonian diffeomorphisms on irrational manifolds.
- Introduces new Floer homology techniques for irrational symplectic manifolds.
- Develops a localization method for filtered Floer homology in short action intervals.

## Abstract

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 over the second fundamental group. In particular, we this removes the rationality condition from similar results. The proof in the irrational case involves several new ideas including the definition and the properties of the filtered Floer homology for Hamiltonians on irrational manifolds. We also develop a method of localizing the filtered Floer homology for short action intervals using a direct sum decomposition, where one of the summands only depends on the behavior of the Hamiltonian in a fixed open set.