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Kolmogorov–Sinai entropy : ウィキペディア英語版
Measure-preserving dynamical system
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular.
==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
:(X, \mathcal, \mu, T)
with the following structure:
*X is a set,
*\mathcal B is a σ-algebra over X,
*\mu:\mathcal\rightarrow() is a probability measure, so that μ(''X'') = 1, and μ(∅) = 0,
* T:X \rightarrow X is a measurable transformation which preserves the measure \mu, i.e., \forall A\in \mathcal\;\; \mu(T^(A))=\mu(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 ∪ , or [0, +∞)), where each transformation ''Ts'' satisfies the same requirements as ''T'' above. In particular, the transformations obey the rules:
* T_0 = id_X :X \rightarrow X, the identity function on ''X'';
* T_ \circ T_ = T_, whenever all the terms are well-defined;
* T_^ = T_, whenever all the terms are well-defined.
The earlier, simpler case fits into this framework by defining''Ts'' = ''Ts'' for ''s'' ∈ N.
The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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