Leafwise fixed points for $C^0$-small Hamiltonian flows

TL;DR

This paper proves the existence of multiple leafwise fixed points for $C^0$-small Hamiltonian flows on coisotropic submanifolds, extending previous results without requiring $C^1$-closeness or contact type assumptions.

Contribution

It establishes the first leafwise fixed point theorem without $C^1$-closeness or contact type conditions, using a $C^0$-smallness assumption on Hamiltonian flows.

Findings

At least the cup-length of $N$ leafwise fixed points exist.

Number of fixed points bounded below by Betti numbers under nondegeneracy.

Nondegeneracy condition is generic and optimal.

Abstract

Consider a closed coisotropic submanifold $N$ of a symplectic manifold $(M,\omega)$ and a Hamiltonian diffeomorphism $\phi$ on $M$. The main result of this article states that $\phi$ has at least the cup-length of $N$ many leafwise fixed points w.r.t. $N$, provided that it is the time-1-map of a global Hamiltonian flow whose restriction to $N$ stays $C^0$-close to the inclusion $N\to M$. If $(\phi,N)$ is suitably nondegenerate then the number of these points is bounded below by the sum of the Betti-numbers of $N$. The nondegeneracy condition is generically satisfied. This appears to be the first leafwise fixed point result in which neither $\phi\big|_N$ is assumed to be $C^1$-close to the inclusion $N\to M$, nor $N$ to be of contact type or regular (i.e., "fibering"). It is optimal in the sense that the $C^0$-condition on $\phi$ cannot be replaced by the assumption that $\phi$ is Hofer-small.