Necessary conditions for stability of diffeomorphisms
Author:
Abstract
S. Smale has recently given sufficient conditions for a diffeomorphism to be Ω \Omega -stable and conjectured the converse of his theorem. The purpose of this paper is to give some limited results in the direction of that converse. We prove that an Ω \Omega -stable diffeomorphism f f has only hyperbolic periodic points and moreover that if p p is a periodic point of period k k then the k k th roots of the eigenvalues of d f p k df_p^k are bounded away from the unit circle. Other results concern the necessity of transversal intersection of stable and unstable manifolds for an Ω \Omega -stable diffeomorphism.
Publisher
American Mathematical Society (AMS)
Subject
Applied Mathematics,General Mathematics
Link
http://www.ams.org/tran/1971-158-02/S0002-9947-1971-0283812-3/S0002-9947-1971-0283812-3.pdf
Reference4 articles.
1. Nongenericity of Ω-stability;Abraham, R.,1970
2. Stable manifolds and hyperbolic sets;Hirsch, Morris W.,1970
3. Differentiable dynamical systems;Smale, S.;Bull. Amer. Math. Soc.,1967
4. The Ω-stability theorem;Smale, S.,1970
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