A Poincar\'e-Birkhoff theorem for Hamiltonian flows on nonconvex domains

TL;DR

This paper extends the Poincaré-Birkhoff theorem to higher-dimensional Hamiltonian systems on nonconvex domains, replacing classical conditions with an avoiding rays condition, broadening the theorem's applicability.

Contribution

It introduces a higher-dimensional version of the Poincaré-Birkhoff theorem applicable to nonconvex domains without requiring maps to be close to identity or monotone.

Findings

Establishes existence of fixed points under new conditions

Generalizes classical theorem to higher dimensions

Applies to nonconvex, non-monotone Hamiltonian systems

Abstract

We present a higher-dimensional version of the Poincar\'e-Birkhoff theorem which applies to Poincar\'e time maps of Hamiltonian systems. The maps under consideration are neither required to be close to the identity nor to have a monotone twist. The annulus is replaced by the product of an $N$-dimensional torus and the interior of a $(N-1)$-dimensional (not necessarily convex) embedded sphere; on the other hand, the classical boundary twist condition is replaced by an avoiding rays condition.