Editing Mixing (mathematics) (section)

=== ''L''<sup>2</sup> formulation ===

The properties of ergodicity, weak mixing and strong mixing of a measure-preserving dynamical system can also be characterized by the average of observables. By von Neumann's ergodic theorem, ergodicity of a dynamical system <math>(X, \mathcal{A}, \mu, T)</math> is equivalent to the property that, for any function <math>f \in L^2 (X, \mu)</math>, the sequence <math>(f \circ T^n)_{n \ge 0}</math> converges strongly and in the sense of Cesàro to {{tmath|1= \int_X f \, d \mu }}, i.e.,
: <math> \lim_{N \to \infty} \left \| {1 \over N} \sum_{n=0}^{N-1} f \circ T^n - \int_X f \, d \mu \right \|_{L^2 (X, \mu)}= 0.</math>

A dynamical system <math>(X, \mathcal{A}, \mu, T)</math> is weakly mixing if, for any functions <math>f</math> and <math>g \in L^2 (X, \mu),</math>
: <math> \lim_{N \to \infty} {1 \over N} \sum_{n=0}^{N-1} \left  | \int_X f \circ T^n \cdot g \, d \mu- \int_X f \, d \mu \cdot \int_X g \, d \mu \right |= 0.</math>

A dynamical system <math>(X, \mathcal{A}, \mu, T)</math> is strongly mixing if, for any function {{tmath|1= f \in L^2 (X, \mu) }}, the sequence <math>(f \circ T^n)_{n \ge 0}</math> converges weakly to {{tmath|1= \int_X f \, d \mu }}, i.e., for any function <math>g \in L^2 (X, \mu),</math>
: <math> \lim_{n \to \infty} \int_X f \circ T^n \cdot g \, d \mu = \int_X f \, d \mu \cdot \int_X g \, d \mu.</math>

Since the system is assumed to be measure preserving, this last line is equivalent to saying that the [[covariance]] {{tmath|1= \lim_{n \to \infty} \operatorname{Cov} (f \circ T^n, g) = 0 }}, so that the random variables <math>f \circ T^n</math> and <math>g</math> become orthogonal as <math>n</math> grows. Actually, since this works for any function {{tmath|1= g }}, one can informally see mixing as the property that the random variables <math>f \circ T^n</math> and <math>g</math> become independent as <math>n</math> grows.