Bifurcation and chain recurrence

1 June 1983

journal article
research article
Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems

Vol. 3 (2) , 231-240
https://doi.org/10.1017/s0143385700001942

Abstract

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.

Keywords

This publication has 1 reference indexed in Scilit:

Attractors: Persistence, and Density of Their Basins
Transactions of the American Mathematical Society, 1982