# 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 . 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 ; for an account of the theory of distal dynamic systems and their generalizations, as well as the relevant literature, see .