Consider a topological dynamical system (tds) $(X, T)$, i.e. a compact metric space $X$ and a continuous map $T : X \to X$ (perhaps a homeomorphism). The ergodic probability measures for $T$ are exactly the extreme points of the simplex $\mathcal{M}_T$ of $T$-invariant Borel probability measures on $X$.
My question: Is there a term for when a tds $(X, T)$ has the property that all ergodic measures for $(X, T)$ also satisfy some stronger ergodicity property (e.g. totally ergodic, weakly mixing, etc.)? Where can I learn more about them?
Here's what I know:
I know such systems exist. For example, take a measure-theoretic Bernoulli system and use Jewett-Krieger to find a uniquely ergodic model. Then the only ergodic measure for that uniquely ergodic model will be the Bernoulli measure. I can't think yet of any non-trivial examples of this phenomenon, though.
On the other hand, if we take an ergodic measure-theoretic dynamical system and use Jewett-Krieger to find a uniquely ergodic model for that system, then that uniquely ergodic model will have exactly one ergodic measure that's exactly as high on the ergodicity hierarchy as the one we started with. So, for example, an irrational rotation on the circle has exactly one ergodic measure (i.e. the Haar measure), but it's not weakly mixing.