Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T04:12:24.998Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

[FW]Furstenberg, H. & Weiss, B.. The finite multipliers of infinite transformation. Lecture Notes in Maths 688 Springer: 1978, pp. 127–132.Google Scholar
[K]Körner, T. W.. Recurrence without uniform recurrence. Ergod. Th. & Dynam. Sys. 7 (1987), 559–566.CrossRefGoogle Scholar
[F]Furstenberg, H.. The structure of distal flows. Amer. J. Math. 85 (1963), 477–515.CrossRefGoogle Scholar
[V]Veech, W. A.. The equicontinuous structure relation for minimal abelian transformation groups. Amer. J. Math. 90 (1968), 723–732.CrossRefGoogle Scholar
[P]Petersen, K. E.. Disjointness and weak mixing of minimal sets. Proc. Amer. Math. Soc. 24 (1970), 278–280.CrossRefGoogle Scholar
[Ke-R]Keynes, H. B. & Robertson, J. B.. Eigenvalue theorems in topological transformation groups. Trans. Amer. Math. Soc. 139 (1969), 359–369.CrossRefGoogle Scholar
[E-Ke]Ellis, R. & Keynes, H.. A characterization of the equicontinuous structure relation. Trans. Amer. Math. Soc. 161 (1971), 171–183.CrossRefGoogle Scholar
[B]Bronstein, I. V.. Extensions of Minimal Transformation Groups. Susthoff and Noordhoff: Alphen aan rijn, 1979.CrossRefGoogle Scholar
[C]Clay, J.. Proximity relation in transformation groups. Trans. Amer. Math. Soc. 108 (1963), 88–96.CrossRefGoogle Scholar
[E]Ellis, R.. Distal transformation groups. Pac. J. Math. 8 (1958), 401–405.CrossRefGoogle Scholar
[Ke]Keynes, H. B.. Lifting of Topological entropy. Proc. Amer. Math. Soc.Google Scholar
[G-W]Glasner, S. & Weiss, B.. On the construction of minimal skew products. Isl. J. Math. 34 (1979), 321–336.CrossRefGoogle Scholar