  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.
