Ergodic group effect
Ergodic group effects allow the use of methods of measure theory and the theory of dynamic systems in group theory in mathematics . Clearly, ergodicity of a group action on a probability space means that almost all points in the probability space lie in a single orbit.
The term generalizes the concepts of ergodic transformation and ergodic flow. One also speaks of ergodic dynamic systems .
definition
Let be a probability measure on a probability space and
a measure preserving effect of a countable group , d. H. for every measurable amount and each should apply.
The group effect is called ergodic if the following applies for every -invariable measurable amount :
- or .
(A lot is -invariant if for also for all follows.)
An equivalent definition says that the effect is ergodic if and only if the only -invariant functions are the constant functions . (A function is called -invariant if the equation holds for- almost all and all .)
Operator theoretical formulation
Denote with the Hilbert space of the square integrable functions , with the algebra of the bounded operators on this Hilbert space and with the ( almost everywhere) bounded functions. Restricted functions act as constrained operators using point multiplication and the elements of the group act as constrained operators using .
Then ergodicity can be defined as follows.
A group effect is ergodic if and only if there is no commuting projection
gives.
Examples
- An invertible ergodic transformation is an ergodic effect of the group .
- An ergodic flow is an ergodic effect of the group .
- A co-compact Klein group has an ergodic effect on the edge at infinity .
literature
- Alexander S. Kechris: Global aspects of ergodic group actions. Mathematical Surveys and Monographs. 160. American Mathematical Society, Providence, RI, 2010, ISBN 978-0-8218-4894-4 .
- Alexander Gorodnik, Amos Nevo: The ergodic theory of lattice subgroups. Annals of Mathematics Studies, 172, Princeton University Press, Princeton, NJ, 2010, ISBN 978-0-691-14185-5 .
- Bachir Bekka, Matthias Mayer: Ergodic theory and topological dynamics of group actions on homogeneous spaces. London Mathematical Society Lecture Note Series, 269, Cambridge University Press, Cambridge, 2000, ISBN 0-521-66030-0 .