# 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 .

## 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).