Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T06:27:27.339Z Has data issue: false hasContentIssue false
  • English
  • Français

Decomposing isometric extensions using group extensions

Published online by Cambridge University Press:  19 September 2008

A. H. Forrest
A. H. Forrest
Affiliation:
DPMMS, Mill Lane, Cambridge, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the structure of isometric extensions of compact metric topological dynamical systems with ℤ action and gives two decompositions of the general case to a more structured case. Suppose that Y → X is a M-isometric extension. An extension, Z, of Y is constructed which is also a G-isometric extension of X, where G is the group of isometries of M. The first construction shows that, provided that (X, T) is transitive, there are almost-automorphic extensions Y′ → Y and X′ → X, so that Y′ is homeomorphic to X′ × M and the natural projection Y′ → X′ is a group extension. The second shows that, provided that (X, T) is minimal, there is a G-action on Z which commutes with T and which preserves fibres and acts on each of them minimally. Each individual orbit closure, Za, in Z is a G′-isometric extension of X, where G′ is a subgroup of G, and there is a G′-action on Za which commutes with T, preserves fibres and acts minimally on each of them. Two illustrations are presented. Of the first: to reprove a result of Furstenberg; that every distal point is IP*-recurrent. Of the second: to describe the minimal subsets in isometric extensions of minimal topological dynamical systems.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 4 , December 1993 , pp. 661 - 673
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ellis, R.. The Veech structure theorem. Trans. Amer. Math. Soc. 186 (1973), 203–218.CrossRefGoogle Scholar
[2]Ellis, R.. Topological dynamics and ergodic theory. Ergod. Th. & Dynam. Sys. 7 (1987), 25–47.CrossRefGoogle Scholar
[3]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press: Princeton, NJ, 1981.CrossRefGoogle Scholar
[4]Furstenberg, H.. The structure of distal flows. Amer. J. Math. 85 (1963), 477–515.CrossRefGoogle Scholar
[5]Hindman, N.. Finite sums from sequences within cells of a partition of IN. J. Combinat. Theory (A)17 (1974), 1–11.CrossRefGoogle Scholar
[6]Loomis, L. H.. An Introduction to Abstract Harmonic Analysis. D. Van Nostrand: New York, 1953.Google Scholar
[7]Veech, W. A.. Point distal flows. Amer. J. Math. 92 (1970), 205–242.CrossRefGoogle Scholar
[8]Zimmer, R. J.. Extensions of ergodic group actions. Ill. J. Math. 20 (1976), 373–409.Google Scholar