# Ergodic theory This item has been on the quality assurance side of the portal mathematics entered. This is done in order to bring the quality of the mathematics articles to an acceptable level . Please help fix the shortcomings in this article and please join the discussion !  ( Enter article )

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 .

## Preparations Example of a (Lebesgue) size-preserving figure: with${\ displaystyle T \ colon [0,1) \ rightarrow [0,1)}$ ${\ displaystyle x \ mapsto 2x \ mod 1}$ 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. ${\ displaystyle (\ Omega, {\ mathcal {A}}, P)}$ ${\ displaystyle T}$ ${\ displaystyle P}$ ${\ displaystyle T}$ ${\ displaystyle P}$ ${\ displaystyle P (T ^ {- 1} (A)) = P (A)}$ ${\ displaystyle A}$ ${\ displaystyle {\ mathcal {A}}}$ ${\ displaystyle (\ Omega, {\ mathcal {A}}, P, T)}$ 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 . ${\ displaystyle A}$ ${\ displaystyle T}$ ${\ displaystyle T ^ {- 1} (A) = A}$ ${\ displaystyle T}$ ${\ displaystyle {\ mathcal {I}}}$ ${\ displaystyle B}$ ${\ displaystyle P}$ ${\ displaystyle P (B \ triangle T ^ {- 1} (B)) = 0}$ ## definition

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. ${\ displaystyle P (A) \ in \ {0; 1 \}}$ ${\ displaystyle (\ Omega, {\ mathcal {A}}, P, T)}$ ${\ displaystyle (\ Omega, {\ mathcal {A}}, P)}$ ${\ displaystyle T}$ 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: ${\ displaystyle (\ Omega, {\ mathcal {A}}, P, T)}$ • ${\ displaystyle (\ Omega, {\ mathcal {A}}, P, T)}$ is ergodic dimensional maintenance system.
• For every quasi-invariant set either or applies .${\ displaystyle A \ in {\ mathcal {A}}}$ ${\ displaystyle P (A) = 0 \,}$ ${\ displaystyle P (A) = 1 \,}$ • Every measurable function is almost certainly constant.${\ displaystyle {\ mathcal {I}}}$ ${\ displaystyle f: \ Omega \ to \ mathbb {R}}$ ${\ displaystyle P}$ • For all true: .${\ displaystyle A, B \ in {\ mathcal {A}}}$ ${\ displaystyle \ lim _ {n \ to \ infty} {\ frac {1} {n}} \ sum _ {k = 0} ^ {n-1} P \ left (A \ cap T ^ {- k} (B) \ right) = P (A) P (B)}$ ## Applications

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 . ${\ displaystyle (\ Omega, {\ mathcal {A}}, P, T)}$ ${\ displaystyle \ Omega = \ mathbb {R} / \ mathbb {Z}}$ ${\ displaystyle {\ mathcal {A}} = {\ mathcal {B}} (\ Omega)}$ ${\ displaystyle P = \ lambda}$ ${\ displaystyle T: \ Omega \ to \ Omega, \; x \ mapsto x + \ alpha {\ bmod {1}}}$ ${\ displaystyle \ alpha \ in \ mathbb {R}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle \ alpha \ in \ mathbb {R} \ setminus \ mathbb {Q}}$ ### 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 ${\ displaystyle 0}$ ${\ displaystyle 1}$ ${\ displaystyle \ Omega = \ {0.1 \} ^ {\ mathbb {N}}}$ ${\ displaystyle {\ mathcal {A}}}$ ${\ displaystyle P}$ ${\ displaystyle P_ {i} (\ {0 \}) = P_ {i} (\ {1 \}) = {\ frac {1} {2}}}$ ${\ displaystyle T}$ ${\ displaystyle \ Omega}$ ${\ displaystyle T}$ ${\ displaystyle T: \ {0.1 \} ^ {\ mathbb {N}} \ to \ {0.1 \} ^ {\ mathbb {N}}, \; T (x) _ {n}: = x_ {n + 1}}$ Then the 4-tuple is an ergodic dynamic system. ${\ displaystyle (\ {0,1 \} ^ {\ mathbb {N}}, {\ mathcal {A}}, P, T)}$ ### Gaussian figure

Let be the base space and the corresponding Borel σ-algebra . The defining Gauss map by ${\ displaystyle \ Omega = [0,1]}$ ${\ displaystyle {\ mathcal {A}} = {\ mathcal {B}} ([0,1])}$ ${\ displaystyle T}$ ${\ displaystyle T: [0,1] \ to [0,1], \; T (x): = {\ begin {cases} {\ tfrac {1} {x}} {\ bmod {1}} & x \ neq 0 \\ 0 & x = 0 \ end {cases}}}$ Now, if a measure the Gaußmaß , is chosen, it is in an ergodic dynamical system. ${\ displaystyle {\ text {v}} (A): = {\ tfrac {1} {\ ln (2)}} \ int _ {A} \, {\ tfrac {1} {1 + x}} \ , \ mathrm {d} \ lambda (x)}$ ${\ displaystyle A \ in {\ mathcal {B}} ([0,1])}$ ${\ displaystyle ([0,1], {\ mathcal {B}} ([0,1]), T, v)}$ ## literature

### Modern

• 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).