No $C^1$-recurrence of iterations of symplectomorphisms

TL;DR

This paper proves that on negatively monotone symplectic manifolds, symplectomorphisms and Hamiltonian diffeomorphisms do not exhibit $C^1$-recurrence, extending previous results and highlighting differences from Hamiltonian group actions.

Contribution

It generalizes the non-$C^1$-recurrence results for symplectomorphisms to negatively monotone manifolds, expanding understanding of their dynamical behavior.

Findings

Symplectomorphisms lack $C^1$-recurrence on negatively monotone manifolds.

Hamiltonian diffeomorphisms also do not have $C^1$-recurrence in this setting.

Negatively monotone symplectic manifolds are distinct from Hamiltonian $G$-manifolds.

Abstract

In this article, we study the behavior of iterations of symplectomorphisms and Hamiltonian diffeomorphisms on symplectic manifolds. We prove that symplectomorphisms and Hamiltonian diffeomorphisms do not have $C^1$-recurrence on negatively monotone symplectic manifolds. This is a generalization of the results of the study of Polterovich, Ono, Atallah-Shelukhin. Hamiltonian group actions play very important roles in symplectic geometry. We see that negatively monotone symplectic manifolds are far from being Hamiltonian $G$-manifolds.