3
$\begingroup$

Let $(X, T):= (\{0,1\}^{\mathbb Z}, \text{the shift transformation})$ be the full two shift system. Then the set $M(X, T)$ consisting of all invariant (probability) measures on that system is a compact metrizable space under the weak* topology so that we can talk about residual subsets and meager subsets of $M(X, T)$.

Now we consider just three kinds of mixing properties: ergodicity, weak-mixing, strong-mixing. My question is whether the set of weak-mixing measures form a residual subset of $M(X, T)$ or a meager subset or otherwise and what is a reference containing the result.

For the other two properties, we know:

  • The ergodic measures form a residual subset. ((21.9) in [DGS])

  • The strong-mixing measures form a meager subset. ((21.13) in [DGS])

[DGS] Ergodic Theory on Compact Spaces - Denker & Grillenberger & Sigmund


Edit: Remarks regarding the answer

The title of the paper referred in the selected answer is "A note on mixing processes". The paper contains a proof of the fact that for the full shift ${\mathbb R}^{\mathbb Z}$ (with the alphabet being the reals), the weakly mixing measures on it form a dense $G_\delta$ set (and hence a residual set). The proof works for the full shift with finite alphabet as well. If I understood the paper right, the weakly mixing measures on a (self-)homeomorphism of a compact metric space always form a G delta set, regardless of whether they are dense or not.

$\endgroup$
2
$\begingroup$

Weak mixing is generic. The result is due to K. R. Parthasarathy, "Indian Journal of Statistics", November 1962, Series A vol.24.

Note that in the measurable setting, this is due to Halmos (see his 1956 book "lectures on ergodic theory").

$\endgroup$
2
$\begingroup$

Weak mixing measures are generic. I do not know of an exact reference for your question. One way to show this is to use the characterisation: $(T,\mu)$ is weak mixing if and only if $(T\times T,\mu\times\mu)$ is ergodic and then modify the proof in [DGS] accordingly. See the paper of Alpern "New proofs that weak mixing is generic" for an application of this method in a related question.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.