Ergodic set

in the phase space $ X $( a metrizable compactum) of a topological dynamical system $ \{ S _ {t} \} $( a flow (continuous-time dynamical system) or a cascade) corresponding to a normalized ergodic invariant measure $ \mu $

A set of points $ x \in X $ such that:

a) for every continuous function $ f : X \rightarrow \mathbf R $ the "time average"

$$ \frac{1}{T} \int\limits _ { 0 } ^ { T } f ( S _ {t} x ) d t \rightarrow \int\limits _ { X } f d \mu \ \textrm{ as } T \rightarrow \infty ; $$

b) $ \mu ( U) > 0 $ for every neighbourhood $ U $ of $ x $.

A point for which the limit of the time average in a) exists for every continuous $ f $ is called quasi-regular. For such a point this limit has the form $ \int f d \mu $, where $ \mu $ is some normalized invariant measure, depending on $ x $ and not necessarily ergodic. If b) holds for this $ \mu $, then the point is called a density point, if this $ \mu $ is ergodic, then the point is called transitive; when both conditions hold, the point is called regular. The set of non-regular points has measure zero relative to any normalized invariant measure. The partition of the set of regular points into ergodic sets corresponding to different $ \mu $ is the strongest realization of the idea of decomposing a dynamical system into ergodic components (which is valid under stronger restrictions on the system).

Ergodic sets were introduced by N.N. Bogolyubov and N.M. Krylov [1]. For other accounts, discussions of various generalizations and related questions see the references to Invariant measure 1) and Metric transitivity.

References

[1]	N. Krylov, N. [N.N. Bogolyubov] Bogoliouboff, "La théorie générale de la mesure dans son application à l'étude des systèmes dynamiques de la mécanique non-linéare" Ann. of Math. Ser. (2) , 38 (1937) pp. 65–113

Comments

A good account of ergodic sets is given in [a1]. Closely related is the notion of a generic point (with respect to a normalized invariant measure $ \mu $): A quasi-regular point such that for every continuous $ f : X \rightarrow \mathbf R $ the limit in a) is $ \int f d \mu $, where $ \mu $ is the given measure.

References

[a1]	J.C. Oxtoby, "Ergodic sets" Bull. Amer. Math. Soc. , 58 (1952) pp. 116–136