Ergodic theory studies the statistical and qualitative behavior of measurable group actions and actions of semigroups
on measure spaces . The time parameters most commonly
range over N , R , R+ ,
or Z .
For a measure-preserving transformation of a probability space , the Poincaré
recurrence theorem guarantees that almost every
point of each measurable set returns to that set infinitely often.
Ergodic theory had its origins in the work of Boltzmann in statistical mechanics problems where time- and space-distribution averages are equal (Moore 2015, p. 1907).
Steinhaus (1999, pp. 237-239) gives a practical application to ergodic theory
to keeping one's feet dry ("in most cases," "stormy weather excepted")
when walking along a shoreline without having to constantly turn one's head to anticipate
incoming waves. The mathematical origins of ergodic theory are due to von Neumann,
Birkhoff, and Koopman in the 1930s (Moore 2015). It has since grown to be a huge
subject and has applications not only to statistical mechanics, but also to number
theory , differential geometry , functional
analysis , etc. There are also many internal problems (e.g., ergodic theory being
applied to ergodic theory) which are interesting.
A smooth positive-volume realization of an invertible measure-preserving system on a probability space , with transformation diffeomorphism compact
manifold preserving a probability measure
with a strictly positive smooth density relative to
the volume measure of a Riemannian metric , and
a measure-preserving bijection sets
of full measure . Both measurable functions ,
and continuous map .
See also Ambrose-Kakutani Theorem ,
Banach's Simple Lebesgue Spectrum
Problem ,
Birkhoff's Ergodic Theorem ,
Dye's Theorem ,
Dynamical
System ,
Ergodic Transformation ,
Hopf's Theorem ,
Invariant
Measure ,
Koopman Operator ,
Mixing
Transformation ,
Multiple Ergodic Average ,
Ornstein's Theorem
Explore with Wolfram|Alpha
References Billingsley, P. Ergodic Theory and Information. Cornfeld, I.;
Fomin, S.; and Sinai, Ya. G. Ergodic
Theory. Katok, A. and Hasselblatt,
B. An
Introduction to the Modern Theory of Dynamical Systems. Moore, C. C. "Ergodic Theorem,
Ergodic Theory, and Statistical Mechanics." Proc. Nat. Acad. Sci. USA 112 ,
1907-1911, 2015. https://doi.org/10.1073/pnas.1421798112 . Nadkarni,
M. G. Basic
Ergodic Theory. Parry, W.
Topics
in Ergodic Theory. Petersen,
K. Ergodic
Theory. Radin,
C. "Ergodic Theory." Ch. 1 in Miles
of Tiles. Sinai,
Ya. G. Topics
in Ergodic Theory. Smorodinsky,
M. Ergodic
Theory, Entropy. Steinhaus,
H. Mathematical
Snapshots, 3rd ed. Walters,
P. Ergodic
Theory: Introductory Lectures. Walters,
P. Introduction
to Ergodic Theory. Referenced
on Wolfram|Alpha Ergodic Theory
Cite this as:
Weisstein, Eric W. "Ergodic Theory." From
MathWorld https://mathworld.wolfram.com/ErgodicTheory.html
Subject classifications
, consists of a 
diffeomorphism 
of a compact
manifold preserving a probability measure
with a strictly positive smooth density relative to
the volume measure of a Riemannian metric, and
a measure-preserving bijection 
satisfying 
on invariant sets
of full measure. Both 
and 
are measurable functions,
and 
is not required to be a continuous map.
