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For measure-preserving dynamical systems, there exist several notions of mixing. The most basic ones are strong mixing, weak mixing and ergodicity (see the wikipedia page, for instance), asserting different degrees of 'decay of correlation' between two arbitrary measurable sets. The weakest of the three is ergodicity, which has proven to be a very useful property. In fact, it allows one to interchange time- and space-averages, which is generally a great thing! I.e., if $T$ is the composition operator of an ergodic transformation, then

\begin{equation*} \frac{f + Tf + \ldots + T^nf}{n+1} \stackrel{\text{a.e.}}{\longrightarrow} \int\limits f ~ d\mu \hskip 10pt \forall f \in L^1. \end{equation*}

There have been many generalisations of Birkhoff's theorem (by E. Hopf; Hurewicz; Chacón and Ornstein; and others). For example, Hurewicz's ergodic theorem implies (in particular) that in the case of ergodicity one has

\begin{equation*} \frac{f + Tf + \ldots + T^nf}{g + Tg + \ldots + T^ng} \stackrel{\text{a.e.}}{\longrightarrow} \frac{\int\limits f ~ d\mu}{\int\limits g ~ d\mu} \hskip 10pt \forall f,g \in L^1, ~ g>0. \end{equation*}

There are more examples (e.g. S. Sawyer's and also E. M. Stein's continuity principle), which suggest that ergodicity is sufficient to obtain 'nice' results in most of the cases. On the other hand, I have not heard of any major theorems in which weak or strong mixing plays an essential role (which is probably due to my lack of knowledge). Yet, there has been a lot of effort in the matter, e.g. in generalising the concept to infinite measure spaces. Hence my question:

Are there good examples illustrating the purpose of (weak or strong) mixing in dynamical systems?

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    $\begingroup$ weak-mixing allows you to conclude that the product of your system with any ergodic system is ergodic. This is sometimes very useful. $\endgroup$
    – Anthony Quas
    Commented May 19, 2014 at 1:05
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    $\begingroup$ For Anosov diffeomorphisms, mixing appears to reduce to a technical assumption for practical purposes, and one that is satisfied in all known cases when the underlying manifold is connected. But there are Anosov flows that are not mixing (though the assumption of mixing still appears to be technical in nature). In particular, for transitive Anosov flows one has the Anosov alternative, which states that an Anosov flow is either a suspension of an Anosov diffeomorphism with constant ceiling or every strong stable and unstable manifold is everywhere dense, in which case it is mixing. $\endgroup$
    – Steve Huntsman
    Commented May 19, 2014 at 14:25
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    $\begingroup$ While I am aware that some important authors disagree with me on this point, I have never been entirely happy with the description of ergodicity as a "mixing property". For example, I would say that an isometry does not in any meaningful sense "mix" different regions of the phase space, but an isometry can nonetheless be ergodic. $\endgroup$
    – Ian Morris
    Commented May 20, 2014 at 10:14
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    $\begingroup$ Thanks @Ian, that is an interesting thought. The original definition [A invariant ==> either A or A^c have measure zero] also stresses that there is a clear distinction between ergodicity and mixing properties. $\endgroup$
    – Henry Wegener
    Commented May 21, 2014 at 14:21

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Furstenberg's proof of Szemerédi's Theorem seems to me a nice example: http://math.stanford.edu/~katznel/24812/bulletin.pdf

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Rates of mixing (also called decay of correlations estimates) are an important ingredient of most proofs of statistical limit laws for dynamical systems, such as central limit laws, the law of the iterated logarithm, almost sure invariance principles and so forth. There are so many papers and authors in this area that I will not even attempt to be comprehensive, but some broadly representative examples of this method include "A vector-valued almost sure invariance principle for hyperbolic dynamical systems" by Melbourne and Nicol, and "Invariance principles for interval maps with an indifferent fixed point" by Pollicott and Sharp.

It is important to note that these arguments rely on a specific rate of mixing for certain functions as opposed to just strong mixing on its own with no quantitative information. I find it harder to think of applications which use exactly the statement that a transformation is strong mixing.

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