  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 $(A,\Gamma)$, the space of (ergodic) $\Gamma$-invariant
states on $A$ is homeomorphic to a subspace of (pure) state space of
$A\rtimes\Gamma$. Various applications of this in topological dynamical systems
and representation theory are obtained. In particular, we prove that the
classification of ergodic $\Gamma$-invariant regular Borel probability measures
on a compact Hausdorff space $X$ is equivalent to the classification a special
type of irreducible representations of $C(X)\rtimes \Gamma$.
