Definition
In the definition of a random dynamical system, one is given a family of maps on a probability space . The measure-preserving dynamical system is known as the base flow of the random dynamical system. The maps are often known as shift maps since they "shift" time. The base flow is often ergodic.
The parameter may be chosen to run over
- (a two-sided continuous-time dynamical system);
- (a one-sided continuous-time dynamical system);
- (a two-sided discrete-time dynamical system);
- (a one-sided discrete-time dynamical system).
Each map is required
- to be a -measurable function: for all,
- to preserve the measure : for all, .
Furthermore, as a family, the maps satisfy the relations
- , the identity function on ;
- for all and for which the three maps in this expression are defined. In particular, if exists.
In other words, the maps form a commutative monoid (in the cases and ) or a commutative group (in the cases and ).
Read more about this topic: Base Flow (random Dynamical Systems)
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