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