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