Feral Curves and Minimal Sets

Abstract: Here we prove that for each Hamiltonian function $H\in \mathcal{C}^\infty(\mathbb{R}^4, \mathbb{R})$ defined on the standard symplectic $(\mathbb{R}^4, \omega_0)$, for which $M:=H^{-1}(0)$ is a non-empty compact regular energy level, the Hamiltonian flow on $M$ is not minimal. That is, we prove there exists a closed invariant subset of the Hamiltonian flow in $M$ that is neither $\emptyset$ nor all of $M$. This answers the four dimensional case of a twenty year old question of Michel Herman, part of which can be regarded as a special case of the Gottschalk Conjecture. Our principal technique is the introduction and development of a new class of pseudoholomorphic curve in the "symplectization" $\mathbb{R} \times M$ of framed Hamiltonian manifolds $(M, \lambda, \omega)$. We call these feral curves because they are allowed to have infinite (so-called) Hofer energy, and hence may limit to invariant sets more general than the finite union of periodic orbits. Standard pseudoholomorphic curve analysis is inapplicable without energy bounds, and thus much of this manuscript is devoted to establishing properties of feral curves, such as area and curvature estimates, energy thresholds, compactness, asymptotic properties, etc.