Periodic Points of Hamiltonian Diffeomorphisms Equal to Nondegenerate Linear Maps at Infinity
Meng Li
TL;DR
This work studies Hamiltonian diffeomorphisms on $\mathbb{R}^{2n}$ that are asymptotically quadratic and, under a twist condition at an isolated homologically nontrivial fixed point, proves the existence of infinitely many simple periodic points with arbitrarily large prime periods when the fixed-point set is finite. The authors develop a refined Floer-theoretic framework, using iteration-compatible Hamiltonians $H^{\times k}$, generating loops $P^{\mu}$, and tailored operations that control the quadratic part at infinity, enabling robust continuation maps between filtered Floer homologies. A key advance is showing that under mild nondegeneracy at infinity, the twist condition propagates through these continuations, leading to the main Theorem 1, which generalizes prior four-dimensional results to higher dimensions without further conditions on the asymptotic quadratic form. They also obtain sharp, low-dimensional refinements (Theorem 2) for specific eigenvalue configurations, employing total and local Floer homology arguments. The results contribute to the broader program connecting fixed-point theory and the Hofer–Zehnder conjecture, with potential applications to asymptotically linear PDEs and Liouville-domain settings.
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.