# On the iterated Hamiltonian Floer homology

## TL;DR

This paper investigates how filtered and local Floer homology of Hamiltonians behave under iteration on symplectically aspherical manifolds, revealing relationships with Euler characteristics, Lefschetz indices, and Smith inequalities.

## Contribution

It establishes new connections between the supertrace of cyclic group actions and classical invariants like Euler characteristics and Lefschetz indices in Floer homology.

## Key findings

- Supertrace equals Euler characteristic for filtered Floer homology.
- Supertrace equals Lefschetz index for local Floer homology.
- Analog of Smith inequality for iterated local homology.

## Abstract

The focus of the paper is the behavior under iterations of the filtered and local Floer homology of a Hamiltonian on a symplectically aspherical manifold. The Floer homology of an iterated Hamiltonian comes with a natural cyclic group action. In the filtered case, we show that the supertrace of a generator of this action is equal to the Euler characteristic of the homology of the un-iterated Hamiltonian. For the local homology the supertrace is the Lefschetz index of the fixed point. We also prove an analog of the classical Smith inequality for the iterated local homology and the equivariant versions of these results.