Haar decompression and amenability of Ellis flows
Let $(X,G)$ be a tame flow and let $K$ be an Ellis group of its enveloping semigroup $E(X,G)$. Although $K$ is a compact Hausdorff topological group in its $\tau$-topology, the inclusion $K\hookrightarrow E(X,G)$ need not be Borel. We show that normalized Haar measure on $K$ nevertheless determines, via the Riesz--Markov theorem, a canonical regular Borel probability measure $\mu_K$ on $E(X,G)$, called its Haar decompression. Haar decompression gives an exact ergodicity description. For an arbitrary flow $(X,G)$, amenability of its Ellis flow is equivalent to hereditary amenability of all finite powers $X^n$. In the tame setting, $(E(X,G),G)$ is amenable if and only if $(X,G)$ is hereditarily amenable. Moreover, for the minimal left ideal $\mathcal{M}$ in $E(X,G)$ containing $K$, $\mu_K$ is $G$-invariant if and only if $(\mathcal{M},G)$ is amenable; when this holds, $\mu_K$ is the unique ergodic measure on $\mathcal{M}$ and $\mathcal{M}=\overline{K}$. Then, we conclude that the ergodic measures of $E(X,G)$ are precisely the Haar decompressions associated with amenable minimal left ideals. We also prove that every minimal tame amenable flow is uniquely ergodic and that every ergodic invariant measure on a tame flow has minimal support. Consequently, every ergodic measure on a tame ambit arises by evaluating a suitable Haar decompression.