Editing Mixing (mathematics) (section)

=== Products of dynamical systems ===

Given two measured dynamical systems <math>(X, \mu, T)</math> and <math>(Y, \nu, S),</math> one can construct a dynamical system <math>(X \times Y, \mu \otimes \nu, T \times S)</math> on the Cartesian product by defining <math>(T \times S) (x,y) = (T(x), S(y)).</math> We then have the following characterizations of weak mixing:<ref name="EinsiedlerWard"> Theorem 2.36, Manfred Einsiedler and Thomas Ward, ''Ergodic theory with a view towards number theory'', (2011) Springer {{isbn|978-0-85729-020-5}}</ref>

: '''Proposition.''' A dynamical system <math>(X, \mu, T)</math> is weakly mixing if and only if, for any ergodic dynamical system {{tmath|1= (Y, \nu, S) }}, the system <math>(X \times Y, \mu \otimes \nu, T \times S)</math> is also ergodic.

: '''Proposition.''' A dynamical system <math>(X, \mu, T)</math> is weakly mixing if and only if <math>(X^2, \mu \otimes \mu, T \times T)</math> is also ergodic. If this is the case, then <math>(X^2, \mu \otimes \mu, T \times T)</math> is also weakly mixing.