Strictly ergodic models for dynamical systems

Benjamin Weiss
1985 Bulletin of the American Mathematical Society  
The action of a group G by homeomorphisms of a compact metric space X is said to be strictly ergodic if there is a unique Borel probability measure JJL fixed by the action, and /i(f/) > 0 for every nonempty open set U C X. For commutative groups G (as well as for general amenable groups) this implies that the action is minimal, since if Xo ^ X is closed and G-invariant there would exist a G-invariant measure supported by Xo which would necessarily be different from fi. Analogously one sees that
more » ... the dynamical system (X, G, fi) must be ergodic. A remarkable result due to R. Jewett [Je] and W. Kreiger [K] says that for G = Z, any ergodic action is isomorphic to a strictly ergodic system. This was extended to G = R by K. Jacobs [Ja] and M. Denker and E. Eberlein [DE]. Thus the topological property of strict ergodicity places no restriction on the measure theoretic properties beyond the obvious ergodicity. It is natural to ask what happens for more general groups G, and what happens, even in the case of Z, when we look at diagrams in the category of ergodic Z-actions rather than simply the objects themselves. In brief our results are: (1) When G is commutative every ergodic action has a strictly ergodic model. (2) Any diagram in the category of ergodic Z-actions with the structure of an inverted tree, i.e., no portion of it looks like has a strictly ergodic model (as a diagram). However, not every measure theoretic triple can have a strictly ergodic model. As a consequence of (2), we can, for example, take any ergodic Z-action that has some point spectrum and provide for it a strictly ergodic model in which all the eigenfunctions are continuous. One can combine (1) and (2) which was formulated for Z-actions for those interested in the classical situation. The method of proof that was developed for (1) is flexible enough to admit further refinements. For example, suppose that G = Z 2 , and the action is given by a pair of commuting transformations T, S which are known to
doi:10.1090/s0273-0979-1985-15399-6 fatcat:iigwom3t7bgvfjs25zs7xceeom