A Poincar\'e--Birkhoff theorem for $C^0$-Hamiltonian maps

TL;DR

This paper extends the Poincaré--Birkhoff theorem to higher dimensions using Floer homology, providing new tools for finding periodic orbits and Hamiltonian chords in complex dynamical systems.

Contribution

It introduces a higher-dimensional Poincaré--Birkhoff theorem via Floer homology and a relative version for Lagrangian submanifolds, advancing Hamiltonian dynamics theory.

Findings

Proves a higher-dimensional Poincaré--Birkhoff theorem using Floer homology.

Establishes a relative version for Lagrangian submanifolds.

Aims to find periodic orbits and Hamiltonian chords in the three-body problem.

Abstract

We prove a higher-dimensional version of the well-known Poincar\'e--Birkhoff theorem, using Floer homology. We also prove a relative version for Lagrangian submanifolds. The motivation is finding periodic orbits and Hamiltonian chords in the circular, restricted three-body problem.