In 1974, Georges Hansel proved that every non-ergodic, aperiodic, invertible
probability measure $\mu$-preserving system has a uniformly ergodic
topological model, meaning that for every continuous function the Cesaro
averages converge uniformly. As a consequence, such a system is a disjoint
union of uniquely ergodic systems.
Moreover, Hansel proves that the model can be made a union of strictly
ergodic (uniquely ergodic and minimal) systems.
With Benjy Weiss, we are currently working on a new proof of Hansel's
result, through which we also add one more property (which we call "purity")
of the topological model:
the strictly ergodic systems cocstituting the model represent only
ergodic measures
from an/a priori /selected set necessary for the ergodic decomposition
of the
initial measure $\mu$. In particular, a pure model has topological entropy
equal to the entropy of $\mu$.