Ergodic theory

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The ergodic theory is a branch of mathematics , which both measure theory and stochastic as well as the theory of dynamic systems is assigned. The origins of the ergodic theory lie in statistical physics . The name is derived from the Greek έργον 'work' and όδος 'way' . For details of the physical term, see ergodicity .


Example of a (Lebesgue) size-preserving figure: with

This is called a probability space a measurable figure maßerhaltend if the size of under again , d. H. for all sets from the σ-algebra . Accordingly, the 4-tuple is called a dimensionally conserving dynamic system.

A set is also called - invariant if it agrees with its archetype, that is , if it holds. The set system of all -invariant sets forms a σ-algebra. Analogously to this, a set is called quasi-invariant if the symmetrical difference of the set with its archetype forms a zero set with regard to the probability measure , i.e. if it applies .


A measure-preserving transformation is now called ergodic if for all T-invariant sets A that . The sets thus form a P-trivial σ-algebra . The 4-tuple consisting of a probability space and an ergodic dimension-preserving mapping is accordingly called an ergodic dynamic system.

In addition to this definition, there are a number of equivalent characterizations. If is a dimensionally maintaining dynamic system, then the following statements are equivalent:

  • is ergodic dimensional maintenance system.
  • For every quasi-invariant set either or applies .
  • Every measurable function is almost certainly constant.
  • For all true: .


From a mathematical point of view, Birkhoff's ergodic theorem represents a variant of the strong law of large numbers for ergodic measurement transformations . It is also possible to consider dependent random variables . The same applies to the L p -Erod theorem .

Examples of ergodic images

Rotation on the unit circle

Consider the system consisting of the set , the Borel σ-algebra , the Lebesgue measure and the map . This system is true to size for everyone . It is also ergodic when it is not rational, i.e. when it applies .

Bernoulli Shift

The Bernoulli shift is also an ergodic mapping: consider the basic space of the - sequences with the associated product σ-algebra and the associated infinite product measure defined by . The Bernoulli mapping is the left shift on the base space , that is, is defined as

Then the 4-tuple is an ergodic dynamic system.

Gaussian figure

Let be the base space and the corresponding Borel σ-algebra . The defining Gauss map by

Now, if a measure the Gaußmaß , is chosen, it is in an ergodic dynamical system.

See also




  • DV Anosov: Ergodic theory. Metric theory of dynamical systems. Springer, accessed on July 30, 2019 .
  • Wladimir Igorewitsch Arnold , André Avez: Ergodic Problems of Classical Mechanics . WA Benjamin, New York 1968 (English).
  • Leo Breiman: Probability . Society for Industrial and Applied Mathematics, 1992, ISBN 0-89871-296-3 , chap. 6 (English, first edition: Addison-Wesley, 1968).
  • Peter Walters: An introduction to ergodic theory . Springer, New York 1982, ISBN 0-387-95152-0 (English).
  • Tim Bedford, Michael Keane, Caroline Series (Eds.): Ergodic theory, symbolic dynamics and hyperbolic spaces . Oxford University Press, 1991, ISBN 0-19-853390-X (English).
  • Joseph M. Rosenblatt, Máté Weirdl: Pointwise ergodic theorems via harmonic analysis . In: Karl E. Petersen, Ibrahim A. Salama (Eds.): Ergodic Theory and its Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference . Cambridge University Press, Cambridge 1995, ISBN 0-521-45999-0 (English).
  • Manfred Einsiedler, Thomas Ward: Ergodic theory with a view towards number theory (=  Graduate Texts in Mathematics . Volume 259 ). Springer London, London 2011, ISBN 978-0-85729-020-5 (English).

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