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?*

anyergodic system is ergodic. This is sometimes very useful. $\endgroup$ – Anthony Quas May 19 '14 at 1:05notmixing (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 May 19 '14 at 14:25