# Ergodic set

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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

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.