Mixing (mathematics) - Mixing in Dynamical Systems

Mixing in Dynamical Systems

A similar definition can be given using the vocabulary of measure-preserving dynamical systems. Let be a dynamical system, with T being the time-evolution or shift operator. The system is said to be strong mixing if, for any, one has

.

For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with replaced by with g being the continuous-time parameter.

To understand the above definition physically, consider a shaker full of an incompressible liquid, which consists of 20% wine and 80% water. If is the region originally occupied by the wine, then, for any part of the shaker, the percentage of wine in after n repetitions of the act of stirring is

In such a situation, one would expect that after the liquid is sufficiently stirred, every part of the shaker will contain approximately 20% wine. This leads to

which implies the above definition of strong mixing.

A dynamical system is said to be weak mixing if one has

\lim_{n\to\infty} \frac {1}{n} \sum_{k=0}^n |\mu (A \cap T^{-k}B) - \mu(A)\mu(B)| = 0.

In other words, is strong mixing if converges towards and weak mixing if this convergence is (at least) in the Cesàro sense.

Weak mixing is a sufficient condition for ergodicity.

For a system that is weak mixing, the shift operator T will have no (non-constant) square-integrable eigenfunctions with associated eigenvalue of one. In general, a shift operator will have a continuous spectrum, and thus will always have eigenfunctions that are generalized functions. However, for the system to be (at least) weak mixing, none of the eigenfunctions with associated eigenvalue of one can be square integrable.

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