Periodic Points of Hamiltonian Diffeomorphisms Equal to Nondegenerate Linear Maps at Infinity

TL;DR

This paper proves that Hamiltonian diffeomorphisms on symplectic Euclidean spaces, which match non-degenerate linear maps at infinity and have an isolated fixed point satisfying the twist condition, possess infinitely many simple periodic points.

Contribution

It establishes the existence of infinitely many simple periodic points under specific conditions, extending understanding of Hamiltonian dynamics at infinity.

Findings

Existence of infinitely many simple periodic points.

Finitely many fixed points imply arbitrarily large prime periods.

Results apply to Hamiltonian diffeomorphisms matching linear maps at infinity.

Abstract

We study Hamiltonian diffeomorphisms on symplectic Euclidean spaces that are equal to non-degenerate linear maps at infinity. Under the assumption that there exists an isolated homologically nontrivial fixed point satisfying the twist condition, we prove the existence of infinitely many simple periodic points. More precisely, if such a diffeomorphism has only finitely many fixed points, then it admits simple periodic points with arbitrarily large prime periods.