Is there an example of a non-trivial measure preserving transformation that is uniquely ergodic and strongly mixing (in the measure theoretic sense)? This was asked here, but with no answer.
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$\begingroup$ How about the trivial system $(X,T)$ with $X=\{0\}$ and $T(0)=0$? $\endgroup$– AlgernonCommented Feb 21, 2016 at 20:50
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$\begingroup$ This kind of examples come naturally from the generalized Ising models. $\endgroup$– Włodzimierz HolsztyńskiCommented Feb 21, 2016 at 21:25
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4$\begingroup$ Any ergodic measure-preserving system is (measurably) isomorphic to a uniquely ergodic system, so in particular, this is true for strongly mixing systems. $\endgroup$– MarkCommented Feb 21, 2016 at 21:40
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$\begingroup$ @Mark Oh, I hadn't realized that. If you put it as an answer, perhaps with some explanation as to why that's true, I'll be happy to accept it! $\endgroup$– George ShakanCommented Feb 21, 2016 at 22:11
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1 Answer
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The Jewett-Krieger Theorem states that every ergodic measure-preserving transformation of a standard probability space can be realised as a uniquely ergodic topological dynamical system on a compact metric space. If the metric entropy of the original system is strictly less than $\log d$, where $d$ is an integer, then additionally this topological dynamical system can be taken to be a closed shift-invariant subset of the full shift on $d$ symbols. A proof of this theorem can be found in the book Ergodic theory on compact spaces by Denker, Grillenberger and Sigmund.
A natural example of a uniquely ergodic mixing flow is the horocycle flow on a surface of constant negative curvature. Explicit examples of mixing uniquely ergodic maps have been constructed by Grillenberger in his paper Constructions of strictly ergodic systems II, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 25 (1972/73) 335-342.
answered Feb 21, 2016 at 23:43