Probability-preserving dynamical system

A probability-preserving dynamical system is a quadruple $(X,\Sigma,\mu,T)$ , where $(X,\Sigma,\mu)$ is a probability space and $T$ is a measure-preserving transformation. The system is invertible when $T$ is invertible modulo null sets.

Iteration

The iterates satisfy $T^0=\mathrm{id}$ and $T^{m+n}=T^mT^n$ for $m,n\geq0$ . Invertibility extends these laws to all integers, giving a measure-preserving action of $\mathbb Z$ .

Statistical questions

Ergodic theory asks how orbit averages, invariant events, and correlations reflect the long-term behavior of the system. The probability measure is part of the data: the same map can be ergodic for one invariant probability measure and nonergodic for another.