Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.

Consider the covariant GNS representation $(H_\phi,\pi_\phi, V_{\phi,\alpha},\xi_\phi)$, together with the selfadjoint projection $E_{\phi,\alpha}$ onto the invariant vectors of $H_\phi$ under the unitary $V_{\phi,\alpha}$.

Are there concrete examples (for commutative and non commutative cases) for which ${\rm dim}(E_{\phi,\alpha}H_\phi)>1$?

Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.

Consider the covariant GNS representation $(H_\phi,\pi_\phi, V_{\phi,\alpha},\xi_\phi)$, together with the selfadjoint projection $E_{\phi,\alpha}$ onto the invariant vectors of $H_\phi$ under the unitary $V_{\phi,\alpha}$.

Are there concrete examples for which ${\rm dim}(E_{\phi,\alpha}H_\phi)>1$?