Distal dynamical system

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 (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 [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 [2]; for an account of the theory of distal dynamic systems and their generalizations, as well as the relevant literature, see [3].

References

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

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) is a trajectory such that the flow is equicontinuous (cf. Equicontinuity) on the orbit closure of this point (cf. [a7]; such a trajectory is also called uniformly almost-periodic, [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 [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 [a1], (15.4) or [3], (3.14.22) for the case that either $X$ is metrizable or $G$ is $\sigma$-compact (see also [a4]) and [a6] for the general case. For yet further generalizations (e.g., to point-distal morphisms, the so-called Veech structure theorem) see [3], (3.15.42), [a3] and [a5].

References

[a1]	R. Ellis, "Lectures on topological dynamics" , Benjamin (1969)
[a2]	R. Ellis, "The Furstenberg structure theorem" Pacific J. Math. , 76 (1978) pp. 345–349
[a3]	R. Ellis, "The Veech structure theorem" Trans. Amer. Math. Soc. , 186 (1973) pp. 203–218
[a4]	E. Ihrig, D. McMahon, "On distal flows of finite codimension" Indian Univ. Math. J. , 33 (1984) pp. 345–351
[a5]	D. McMahon, L.J. Nachman, "An instrinsic characterization for PI-flows" Pacific J. Math. , 89 (1980) pp. 391–403
[a6]	D. McMahon, T.S. Wu, "Distal homomorphisms of non-metric minimal flows" Proc. Amer. Math. Soc. , 82 (1981) pp. 283–287
[a7]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)