Generic transitivity for couples of Hamiltonians

TL;DR

This paper demonstrates that for generic pairs of Hamiltonians on a symplectic manifold, the orbits and reachable sets are typically the entire manifold, highlighting a strong form of transitivity in Hamiltonian dynamics.

Contribution

It establishes that generically, pairs of Hamiltonians produce orbits and reachable sets that cover the whole manifold, using a novel strong genericity framework.

Findings

Orbits of generic Hamiltonian pairs are the entire manifold.

Reachable sets are full for generic Hamiltonian pairs on compact manifolds.

Results apply to a strong notion of genericity involving rectifiable subsets.

Abstract

We study orbits and reachable sets of generic couples of Hamiltonians $H_1, H_2$ on a symplectic manifold $N$. We prove that, $C^k$-generically for $k$ large enough, orbits coincide with the whole of $N$, and that the same is true for reachable sets when $N$ is compact. Our results are stated in terms of a strong form of genericity which makes use of the notion of rectifiable subsets of positive codimension in Banach or Frechet spaces.