Ergodic measure

The ergodic measure is a term from probability theory and the theory of dynamic systems . Clearly, ergodicity of a measure with respect to a mapping means that almost all points of the probability space lie in a single orbit of the dynamic system.

definition

Let it be a probability measure on a probability space and a measure-preserving map . Then is an ergodic measure if for every -invariant set : ${\ displaystyle \ mu}$ ${\ displaystyle \ Omega}$ ${\ displaystyle T \ colon \ Omega \ to \ Omega}$ ${\ displaystyle \ mu}$ ${\ displaystyle T}$ ${\ displaystyle A \ subset \ Omega}$

{\ displaystyle \ mu (A) = 0}

or .

{\ displaystyle \ mu (A) = 1}

(A set is called -invariant if it holds, where the archetype of below denotes.) ${\ displaystyle A}$ ${\ displaystyle T}$ ${\ displaystyle T ^ {- 1} A = A}$ ${\ displaystyle T ^ {- 1} A}$ ${\ displaystyle A}$ ${\ displaystyle T}$

existence

It is a compact room . Then the set of -invariant measures is not empty and one can prove that the ergodic measures are the extreme points of the compact, convex metric space . In particular, there are ergodic measures. ${\ displaystyle \ Omega}$ ${\ displaystyle {\ mathcal {M}}}$ ${\ displaystyle T}$ ${\ displaystyle {\ mathcal {M}}}$

Ergo decomposition

If there is only one ergodic measure for an image, then it is clearly called ergodic. (In particular, there is then only one invariant measure.) In general there are several ergodic measures for a given mapping and every invariant measure can be decomposed as a convex combination of ergodic measures using the ergodic decomposition.

literature

Vladimir Abramovich Rokhlin: On the fundamental ideas of measure theory. In: Matematicheskii Sbornik. 67, No. 1, 1949, pp. 107-150 (Russian, abstract ).
PR Halmos: Lectures on Ergodic Theory. Chelsea, New York 1956.
P. Walters: An Introduction to Ergodic Theory. Springer, New York 1982, ISBN 0-387-90599-5 .
A. Katok, B. Hasselblatt: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995. ISBN 0-521-34187-6 .

Web links

Ergodic Measure (MathWorld)