Topological mixing and uniquely ergodic systems
Let \(X\) be a compact space and let \(T\) be a homeomorphism of \(X\) into itself. The dynamical system \((X,T)\) is called uniquely ergodic if \(T\) leaves invariant precisely one regular probability Borel measure. \textit{R. Jewett} [J. Math. Mech. 19, 717-729 (1970; Zbl 0192.406)] proved that every invertible weakly mixing measure preserving transformation of a Lebesgue space \((X,T,{\mathcal A},\lambda)\) is isomorphic to a uniquely ergodic dynamical system \((Y,U,{\mathcal B},\mu).\) It follows from a result of \textit{G. Hansel} and \textit{J. P. Raoult} [Indiana Univ. Math. J. 23, 221-237 (1973; Zbl 0275.28017)] that this assertion remains true replacing ``weakly mixing'' by ``ergodic''. Under the same assumptions it is proved in the paper under review that \((X,T,{\mathcal A},\lambda)\) is isomorphic to a uniquely ergodic topologically mixing dynamical system \((Y,U,{\mathcal B},\mu).\) The proof relies essentially on the technique of uniform partitions developed by Hansel and Raoult (loc.cit.).
weakly mixing measure preserving transformation
scientific article; zbMATH DE number 4034055