Measure-preserving transformation

In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular.

A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system

with the following structure:

This definition can be generalized to the case in which T is not a single transformation that is iterated to give the dynamics of the system, but instead is a monoid (or even a group) of transformations T_s : X → X parametrized by s ∈ Z (or R, or N ∪ {0}, or [0, +∞)), where each transformation T_s satisfies the same requirements as T above. In particular, the transformations obey the rules:

The earlier, simpler case fits into this framework by definingT_s = T^s for s ∈ N.

The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem.

Examples include:

The concept of a homomorphism and an isomorphism may be defined.

Consider two dynamical systems ${\displaystyle (X,{\mathcal {A}},\mu ,T)}$ and ${\displaystyle (Y,{\mathcal {B}},\nu ,S)}$. Then a mapping

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