Flows on one-dimensional spaces. (English) Zbl 0677.54032

Aarts, J. M.; Martens, M.

Fundam. Math. 131, No. 1, 53-67 (1988).

The problem of the paper is to determine on what one-dimensional spaces one can have a flow. To exclude the trivial case where all points of the flow are fixed the authors consider only flows without a rest point. Certain suspensions provide examples of one-dimensional flows with a fixed point. The following converse is proved. Let \(\pi\) : \(X\times R\to X\) be a dynamical system witout rest points on a one-dimensional space X. Then, there exists a zero-dimensional space S and a homeomorphism f: \(S\to S\) such that (\(\pi\),X,R) is topologically equivalent to the suspension \(\Sigma\) (S,f).
Let X be a separable, metric space. X is said to be a matchbox manifold of for each \(x\in X\) there is a zero-dimensional space \(S_ x\) such that \(S_ x\times R\) is homeomorphic to an open neighborhood of x. The authors show that if \(\pi\) : \(X\times R\to X\) is any flow on a one- dimensional space X and if \(\pi\) has no fixed points, then X is a matchbox manifold.

Reviewer: W.R.Utz

MSC:

54H20

Topological dynamics (MSC2010)

Keywords:

one-dimensional flows; suspension; zero-dimensional space; fixed points; matchbox manifold