Rigidity in topological dynamics. (English) Zbl 0661.58027

Glasner, S.; Maon, D.

Ergodic Theory Dyn. Syst. 9, No. 2, 309-320 (1989).

In analogy with the ergodic theoretical notion, we introduce notions of rigidity for a minimal flow (X,T) according to the various ways a sequence \(T^{n_ i}\) can tend to the identity transformation. The main results obtained are: (i) On a rigid flow there exists a T-invariant, symmetric, closed relation \(\bar N\) such that (X,T) is uniformly rigid iff \(\bar N=\Delta\), the diagonal relation. (ii) For syndetically distal (hence distal) flows rigidity is equivalent to uniform rigidity. (iii) We construct a family of rigid flows which includes Körner’s example, in which \(\bar N\) exhibits kinds of behaviour, e.g. \(\bar N\) need not be an equivalence relation. (iv) The structure of flows in the above mentioned family is investigated. It is shown that these flows are almost automorphic.

Reviewer: S.Glasner

MSC:

37C10

Dynamics induced by flows and semiflows

Keywords:

minimal flow; rigid flow; distal; almost automorphic; topological rigidity

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