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I'm looking for an example of a dynamical system which is both (measure-theoretically) strongly mixing and uniquely ergodic. I've looked around and found lots of discussion of systems which are uniquely ergodic but not strongly mixing, but which doesn't seem to think there's any tension between the properties, so I suspect there's an easy answer I just don't know.

Henry
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    In the continuous time setting, horocycle flows on compact surfaces with constant negative curvature provide interesting examples of uniquely ergodic dynamical systems which are strongly mixing. I have no answer regarding to the discrete time case. – Ahriman Feb 20 '15 at 14:21
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    Now cross-posted here: http://mathoverflow.net/questions/231776/ – YCor Feb 21 '16 at 21:03

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Not sure whether you mean topologically or measure theoretically strongly mixing. In the first case the following paper might provide the example you are looking for: http://www.ams.org/journals/tran/1970-148-02/S0002-9947-1970-0259884-8/

but probably your question is about measure theoretic strong mixing and there usually you get many ergodic measures (as most of such systems have periodic points).

MHS
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  • I do mean measure theoretically strongly mixing. Surely it's either clear in the literature that 1) there are such spaces, 2) there provably aren't any such spaces, or 3) the existence of such a space is an interesting open question, but I can't figure out which of these is the case. – Henry Feb 03 '15 at 12:51