  Motivated by Furstenberg's Theorem on sets in the circle invariant under
multiplication by a non-lacunary semigroup, we define a general class of
dynamical systems possessing similar topological dynamical properties. We call
such systems chaotic almost minimal, reflecting that these systems are chaotic,
but in some sense are close to minimal. We study properties of the acting group
needed to admit such an action, and show the existence of a chaotic almost
minimal $\mathbb{Z}$-action. We show there exists chaotic almost minimal
$\mathbb{Z}^{d}$-actions which support multiple distinct nonatomic ergodic
probability measures.
