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Real-expansive flows and topological dimension

Published online by Cambridge University Press:  19 September 2008

H. B. Keynes  and
M. Sears
H. B. Keynes*
Affiliation:
From the School of Mathematics, University of Minnesota, USA;
M. Sears
Affiliation:
Department of Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa
*
†Address for correspondence: Dr H. B. Keynes, School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA.
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Abstract

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We examine generalizations of R. Mañé's results on the topological dimension of spaces supporting an expansive homeomorphism to the case of real-expansive flows. We show that a space supporting a real-expansive flow must be finite dimensional, and a minimal real-expansive flow not exhibiting a type of spiral behaviour must be one-dimensional. This latter class includes all known examples and a slight generalization of Axiom A flows. These results are obtained by introducing a new concept of stable and unstable sets for real flows, and examining real-expansive flows in terms of these sets.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 1 , Issue 2 , June 1981 , pp. 179 - 195
Copyright
Copyright © Cambridge University Press 1981

References

REFERENCES

[1]Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 1–30.CrossRefGoogle Scholar
[2]Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (2) (1973), 429–460.CrossRefGoogle Scholar
[3]Bowen, R. & Walters, P.. Expansive one-parameter flows. J. of Diff. Equations 12 (1972), 180–193.CrossRefGoogle Scholar
[4]Keynes, H. B. & Sears, M.. Modelling expansion in real flows. Pacific J. of Math. 85 (1979), 111–124.CrossRefGoogle Scholar
[5]Mañé, R.. Expansive homeomorphisms and topological dimension. Trans. AMS 252 (1979), 313–319.CrossRefGoogle Scholar
[6]Whitney, H.. Regular families of curves. Ann. of Math. 34 (1933), 244–270.CrossRefGoogle Scholar