Editing Mixing (mathematics) (section)

=== Examples ===

[[Irrational rotation]]s of the circle, and more generally irreducible translations on a torus, are ergodic but neither strongly nor weakly mixing with respect to the Lebesgue measure.

Many maps considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the [[dyadic map]], [[Arnold's cat map]], [[horseshoe map]]s, [[Kolmogorov automorphism]]s, and the [[Anosov flow]] (the [[geodesic flow]] on the unit [[tangent bundle]] of [[compact manifold]]s of [[negative curvature]].)

The dyadic map is "shift to left in binary". In general, for any <math>n \in \{2, 3, \dots\}</math>, the "shift to left in base {{tmath|1= n }}" map <math>T(x) = nx \bmod 1</math> is strongly mixing on the covering family {{tmath|1= \left\{ \left( \tfrac{k}{n^s}, \tfrac{k+1}{n^s} \right) \smallsetminus \Q: s \geq 0, \leq k < n^s \right\} }}, therefore it is strongly mixing on {{tmath|1= (0, 1) \smallsetminus \Q }}, and therefore it is strongly mixing on {{tmath|1= [0, 1] }}.

Similarly, for any finite or countable alphabet {{tmath|1= \Sigma }}, we can impose a discrete probability distribution on it, then consider the probability distribution on the "coin flip" space, where each "coin flip" can take results from {{tmath|1= \Sigma }}. We can either construct the singly-infinite space <math>\Sigma^\N</math> or the doubly-infinite space {{tmath|1= \Sigma^\Z }}. In both cases, the '''shift map''' (one letter to the left) is strongly mixing, since it is strongly mixing on the covering family of cylinder sets. The [[Baker's map]] is isomorphic to a shift map, so it is strongly mixing.