Let A be a $C^*$-algebra with unit $I$, and G a locally compact (Hausdorff) group. An action $\alpha$ of G on A is a strongly continuous homomorphism of G into Aut(A), the group of *-automorphisms of A. We say that $\alpha$ is ergodic if the elements $\lambda I$, with $\lambda$ any complex number are the only elements invariant under $\alpha$.

If A is commutative, i.e. $A\cong C(X)$ where $X$ is a compact Hausdorff space (the spectrum of A), giving an action $\alpha$ on A is equivalent to giving an action $\phi$ of $G$ on the spectrum by homeomorphisms, where for any $f\in C(X)$ we have $$(\alpha_g (f))=f(\phi_g^{-1}(x))\quad \forall x\in X$$

(1) is the strong continuity of $g\mapsto\alpha_g$ equivalent to continuity of the map $g\mapsto\phi_g$ if we endow the set of homeomorphisms:$X\rightarrow X$ with the compact-open topology?

It is easy that under the assumption $G$ compact, ergodicity is equivalent to transitivity of the action $\phi$ on the spectrum X (using the averaging operator), but

(2) what can we say about ergodicity in the case where $G$ is just locally compact (and Hausdorff)? Is ergodicity of $\alpha$ equivalent to 'topological' ergodicity, i.e. the only open (equivalently closed) subsets invariant for $\phi$ are X and $\emptyset$

If we have non-constant invariant functions on $X$, then the pre-image of a value taken is a proper invariant closed of $X$, but what about the other implication?

Thank you in advance