# 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 (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)

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].