What good is (strong) mixing in dynamical systems? 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?
 A: 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.
A: Furstenberg's proof of Szemerédi's Theorem seems to me a nice example:
http://math.stanford.edu/~katznel/24812/bulletin.pdf
