View source for Distal dynamical system

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A dynamical system $\{T^t\}$ with a metric phase space $X$ such that for any points $x\neq y$ the greatest lower bound of the distances,

$$\inf_t\rho(T^tx,T^ty)>0.$$

If a pair of points $x\neq y$ in a given dynamical system has this property, one says that this pair of points is distal; thus, a distal dynamical system is a dynamical system all pairs of points $x\neq y$ of which are distal.

This definition is suitable for  "general"  dynamical systems, when the  "time"  $t$ runs through an arbitrary group $G$. Interesting results are obtained if $G$ is locally compact (the  "classical"  cases of a cascade or flow, viz. when $G=\mathbf Z$ or $G=\mathbf R$, are fundamental, but their treatment is hardly simpler), and $X$ is compact. Of special interest is the case when $X$ is a [[Minimal set|minimal set]] (the general case is reducible, in a sense, to this case, since under the above restriction (the closure of) each trajectory is a minimal set). The most important example of a distal dynamical system is the system resulting from the closure of an almost-periodic trajectory of some dynamical system. A second example are nil-flows [[#References|[1]]]. As is also the case in the above examples, the construction of a distal dynamical system with a minimal set $X$ under these conditions permits a fairly detailed description of an algebraic nature [[#References|[2]]]; for an account of the theory of distal dynamic systems and their generalizations, as well as the relevant literature, see [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Auslander,   L. Green,   F. Hahn,   "Flows on homogeneous spaces" , Princeton Univ. Press  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Furstenberg,   "The structure of distal flows"  ''Amer. J. Math.'' , '''85''' :  3  (1963)  pp. 477–515</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.U. Bronshtein,   "Extensions of minimal transformation groups" , Sijthoff &amp; Noordhoff  (1979)  (Translated from Russian)</TD></TR></table>



====Comments====
There are several notions of  "almost-periodic trajectory"  in use. In the article above, an almost-periodic trajectory of a point in a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] is a trajectory such that the flow is equicontinuous (cf. [[Equicontinuity|Equicontinuity]]) on the orbit closure of this point (cf. [[#References|[a7]]]; such a trajectory is also called uniformly almost-periodic, [[#References|[3]]]).

The Furstenberg structure theorem referred to above was proved originally for any distal minimal dynamical system on a compact metric space $X$ and arbitrary group $G$. In [[#References|[a2]]], the condition that $X$ is metrizable was removed. There is also a so-called  "relative"  version of Furstenberg's theorem, applicable to distal morphisms between compact minimal dynamical systems: see [[#References|[a1]]], (15.4) or [[#References|[3]]], (3.14.22) for the case that either $X$ is metrizable or $G$ is $\sigma$-compact (see also [[#References|[a4]]]) and [[#References|[a6]]] for the general case. For yet further generalizations (e.g., to point-distal morphisms, the so-called Veech structure theorem) see [[#References|[3]]], (3.15.42), [[#References|[a3]]] and [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Ellis,   "Lectures on topological dynamics" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Ellis,   "The Furstenberg structure theorem"  ''Pacific J. Math.'' , '''76'''  (1978)  pp. 345–349</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Ellis,   "The Veech structure theorem"  ''Trans. Amer. Math. Soc.'' , '''186'''  (1973)  pp. 203–218</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Ihrig,   D. McMahon,   "On distal flows of finite codimension"  ''Indian Univ. Math. J.'' , '''33'''  (1984)  pp. 345–351</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. McMahon,   L.J. Nachman,   "An instrinsic characterization for PI-flows"  ''Pacific J. Math.'' , '''89'''  (1980)  pp. 391–403</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D. McMahon,   T.S. Wu,   "Distal homomorphisms of non-metric minimal flows"  ''Proc. Amer. Math. Soc.'' , '''82'''  (1981)  pp. 283–287</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>