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Bifurcation and chain recurrence

Published online by Cambridge University Press:  19 September 2008

M. Hurley
M. Hurley
Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106, U.S.A.
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Abstract

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We show that there is a residual subset of the set of C1 diffeomorphisms on any compact manifold at which the map

is continuous. As this number is apt to be infinite, we prove a localized version, which allows one to conclude that if f is in this residual set and X is an isolated chain component for f, then

(i) there is a neighbourhood U of X which isolates it from the rest of the chain recurrent set of f, and

(ii) all g sufficiently C1 close to f have precisely one chain component in U, and these chain components approach X as g approaches f.

(ii) is interpreted as a generic non-bifurcation result for this type of invariant set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 2 , June 1983 , pp. 231 - 240
Copyright
Copyright © Cambridge University Press 1983

References

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