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Rigidity in topological dynamics

Published online by Cambridge University Press:  19 September 2008

S. Glasner  and
D. Maon
S. Glasner
Affiliation:
School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
D. Maon
Affiliation:
School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
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Abstract

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By 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 2 , June 1989 , pp. 309 - 320
Copyright
Copyright © Cambridge University Press 1989

References

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