Given a topological dynamical system $(X,T)$ (so that $T$ is a homeomorphism of the compact metric space $X$) and a point $x\in X$ we call the set ${\mathcal O}(x):=\overline{\{T^nx:n\in\mathbb Z\}}$ the orbit closure of $x$.

Question 0: Is there a name for systems with the property that the orbit closure of every point is uniquely ergodic (i.e., supports a unique invariant measure)?

It is well known that nilsystems have this property but not all distal systems do.

Question 1: Is it true that if $(X,T)$ and $(Y,S)$ are uniquely ergodic, then $(X\times Y,T\times S)$ has the property that every orbit closure is uniquely ergodic?

Question 2: What if in addition $(X,T)$ and $(Y,S)$ are distal?


1 Answer 1


I think the two-sided Morse system is a counterexample to both questions the first question. Consider the two points $x^{(1)}=\ldots 0110100110010110\cdot0110100110010110\ldots$ and $x^{(2)}=\ldots 1001011001101001\cdot0110100110010110$. The orbit closure of $(x^{(1)},x^{(2)})$ supports exactly two ergodic invariant measures, the diagonal measure on the Morse system and the anti-diagonal measure.

By the way, I have a paper with Emmanuel Lesigne and Máté Wierdl establishing lots of strange properties of the Morse system (such as for any point, $x=(x^{(1)},x^{(2)},x^{(3)})$ of $M^3$, $x$ is generic for some ergodic invariant measure, but there exist points of $M^4$ that are not generic for any invariant measure).

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    $\begingroup$ Thanks for the example and the reference. This does answer the first question, but I'm not sure the Morse system is distal. In fact I thought no symbolic system can be distal (unless it is finite). But perhaps one can project onto the largest distal factor and still have the same property? $\endgroup$ Jul 11, 2017 at 9:45
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    $\begingroup$ Oops. I'm not so used to thinking about distal systems. The lack of distality is shown exactly by the two points that I use (and essentially no other points). I suspect the maximal distal factor is just the odometer, which, of course, does have the property you are asking about. $\endgroup$ Jul 11, 2017 at 14:17

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