Rigidity in topological dynamics

Abstract

AbstractBy analogy with the ergodic theoretical notion, we introduce notions of rigidity for a minimal flow (X, T) according to the various ways a sequence Tni can tend to the identity transformation. The main results obtained are:(i) On a rigid flow there exists a T-invariant, symmetric, closed relation Ñ such that (X, T) is uniformly rigid iff Ñ = Δ, 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 Ñ exhibits various kinds of behaviour, e.g. Ñ 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.