topological size of the set of weakly mixing measures on the full two-shift

Question

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

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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.

Answer 1

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").

Answer 2

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.