View source for Centre of a topological dynamical system

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c0212801.png" /> (of a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] or of a [[Cascade|cascade]]) with phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c0212802.png" />''

The largest closed [[Invariant set|invariant set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c0212803.png" /> for which all points are non-wandering (cf. [[Non-wandering point|Non-wandering point]]) under the restriction of the original system to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c0212804.png" />. (Cf. also [[Topological dynamical system|Topological dynamical system]].) The centre is necessarily non-empty if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c0212805.png" /> 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 [[#References|[1]]]–[[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c0212806.png" /> (see [[#References|[5]]], [[#References|[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 [[#References|[7]]]), but it can be an arbitrarily large countable transfinite number even for flows in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c0212807.png" /> and on certain open surfaces (see [[#References|[8]]]–[[#References|[10]]]). In a complete metric space the centre coincides with the closure of the set of points having the property of [[Poisson stability|Poisson stability]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c0212808.png" /> is compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c0212809.png" /> is a neighbourhood of the centre, then the trajectory of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c02128010.png" />  "stays most of the time in U" : The fraction in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c02128011.png" /> of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c02128012.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c02128013.png" /> tends to 1 as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021280/c02128014.png" />. However, the smallest closed invariant set having this property (the minimal centre of attraction, see [[#References|[3]]] and [[#References|[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|Ergodic set]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.D. Birkhoff,   "Dynamical systems" , Amer. Math. Soc.  (1927)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K.S. Sibirskii,   "Introduction to topological dynamics" , Noordhoff  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D.A. Neumann,   "Central sequences in flows on 2-manifolds of finite genus"  ''Proc. Amer. Math. Soc.'' , '''61''' :  1  (1976)  pp. 39–43</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.G. Maier,   "On central trajectories and a problem of Birkhoff"  ''Mat. Sb.'' , '''26''' :  2  (1950)  pp. 265–290  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D.A. Neumann,   "Central sequences in dynamical systems"  ''Amer. J. Math.'' , '''100''' :  1  (1978)  pp. 1–18</TD></TR></table>