Definition
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:
- X is a set,
- is a σ-algebra over X,
- is a probability measure, so that μ(X) = 1, and μ(∅) = 0,
- T : X → X is a measurable transformation which preserves the measure μ, i. e. each satisfies μ(T−1A) = μ(A).
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 Ts : X → X parametrized by s ∈ Z (or R, or N ∪ {0}, or [0, +∞)), where each transformation Ts satisfies the same requirements as T above. In particular, the transformations obey the rules
- T0 = idX : X → X, the identity function on X;
- , whenever all the terms are well-defined;
- , whenever all the terms are well-defined.
The earlier, simpler case fits into this framework by definingTs = Ts for s ∈ N.
The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem.
Read more about this topic: Measure-preserving Dynamical System
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