Centre of a topological dynamical system

$\{S_t\}$ (of a flow (continuous-time dynamical system) or of a cascade) with phase space $X$

The largest closed invariant set $A\subset X$ for which all points are non-wandering (cf. Non-wandering point) under the restriction of the original system to $A$. (Cf. also Topological dynamical system.) The centre is necessarily non-empty if the space $X$ is compact (more generally, if there is a semi-trajectory with compact closure). G.D. Birkhoff, who introduced the concept of a centre, used another, but equivalent, definition by means of a certain transfinite process (see [1]–[4]). The number of steps of this process is called the depth of the centre (in fact, there are several "depths", since this process allows certain modifications). The depth of the centre is not large for flows on compact manifolds of dimension $\leq2$ (see [5], [6]) or for cascades obtained by the iteration of a homeomorphism of a circle or of an (even non-invertible) continuous mapping of an interval (see [7]), but it can be an arbitrarily large countable transfinite number even for flows in $\mathbf R^3$ and on certain open surfaces (see [8]–[10]). In a complete metric space the centre coincides with the closure of the set of points having the property of Poisson stability.

If $X$ is compact and $U$ is a neighbourhood of the centre, then the trajectory of each point $x\in X$ "stays most of the time in U": The fraction in $[0,T]$ of those $t$ for which $S_tx\in U$ tends to 1 as $T\to\infty$. However, the smallest closed invariant set having this property (the minimal centre of attraction, see [3] and [4]) is, in general, only a part of the centre; in the metrizable case it coincides with the closure of the union of all ergodic sets (cf. Ergodic set).

References

[1]	G.D. Birkhoff, "Ueber gewisse Zentralbewegungen dynamischer Systeme" Nachr. Gesells. Wiss. Göttingen Math. Phys. Kl. : 1 (1926) pp. 81–92 (Collected Math. Papers, Vol. II, pp. 283–294)
[2]	G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927)
[3]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[4]	K.S. Sibirskii, "Introduction to topological dynamics" , Noordhoff (1975) (Translated from Russian)
[5]	A.F. Schwartz, E.S. Thomas, "The depth of the centre of 2-manifolds" , Global stability , Proc. Symp. Pure Math. , 14 , Amer. Math. Soc. (1970) pp. 253–264
[6]	D.A. Neumann, "Central sequences in flows on 2-manifolds of finite genus" Proc. Amer. Math. Soc. , 61 : 1 (1976) pp. 39–43
[7]	O.M. Sharkov'skii, "Fixed points and the centre of a continuous mapping of the line into itself" Dopov. Akad. Nauk. Ukr.RSR , 7 (1964) pp. 865–868 (In Russian) (English summary)
[8]	A.G. Maier, "On central trajectories and a problem of Birkhoff" Mat. Sb. , 26 : 2 (1950) pp. 265–290 (In Russian)
[9]	L.P. Shil'nikov, "On the work of A.G. Maier on central motions" Math. Notes , 5 : 3 (1969) pp. 204–206 Mat. Zametki , 5 : 3 (1969) pp. 335–339
[10]	D.A. Neumann, "Central sequences in dynamical systems" Amer. J. Math. , 100 : 1 (1978) pp. 1–18