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Ergodic invariant states and irreducible representations of crossed product $C^*$-algebras

Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,Γ)$, the space of (ergodic) $Γ$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimesΓ$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $Γ$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)\rtimes Γ$.