Necessary conditions for stability of diffeomorphisms

Franks, John

doi:10.1090/S0002-9947-1971-0283812-3

Necessary conditions for stability of diffeomorphisms
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Trans. Amer. Math. Soc. 158 (1971), 301-308 Request permission

Abstract:

S. Smale has recently given sufficient conditions for a diffeomorphism to be $\Omega$-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 $\Omega$-stable diffeomorphism $f$ has only hyperbolic periodic points and moreover that if $p$ is a periodic point of period $k$ then the $k$th roots of the eigenvalues of $df_p^k$ are bounded away from the unit circle. Other results concern the necessity of transversal intersection of stable and unstable manifolds for an $\Omega$-stable diffeomorphism.

References

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Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR 0271991
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
S. Smale, The $\Omega$-stability theorem, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 289–297. MR 0271971