Necessary conditions for stability of diffeomorphisms

Author:
John Franks

Journal:
Trans. Amer. Math. Soc. **158** (1971), 301-308

MSC:
Primary 57.20

DOI:
https://doi.org/10.1090/S0002-9947-1971-0283812-3

MathSciNet review:
0283812

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

**[1]**R. Abraham and S. Smale,*Nongenericity of Ω-stability*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 5–8. MR**0271986****[2]**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****[3]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math. Soc.**73**(1967), 747–817. MR**0228014**, https://doi.org/10.1090/S0002-9904-1967-11798-1**[4]**S. Smale,*The Ω-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**

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DOI:
https://doi.org/10.1090/S0002-9947-1971-0283812-3

Keywords:
-stability,
structural stability,
hyperbolic structure,
nonwandering set

Article copyright:
© Copyright 1971
American Mathematical Society