Chaotic almost minimal actions

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.