Necessary conditions for stability of diffeomorphisms

TLDR: In this article, it was shown that an in-stable diffeomorphism has only hyperbolic periodic points and if p is a periodic point of period k then the Arth roots of the eigenvalues of dff are bounded away from the unit circle.

Abstract: S Smale has recently given sufficient conditions for a diffeomorphism to be Q-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 in- stable diffeomorphism / has only hyperbolic periodic points and moreover that if p is a periodic point of period k then the Arth roots of the eigenvalues of dff are bounded away from the unit circle Other results concern the necessity of transversal intersection of stable and unstable manifolds for an fi-stable diffeomorphism Introduction We will say that a diffeomorphism /: M—s- M of a compact manifold is Cl-stable if (a) there is a neighborhood N(f) of/in the C1 topology such that g e N(f) implies there is a homeomorphism « from the nonwandering set of/, Q(f) to the nonwandering set of g, 0(g) which satisfies g-« = «•/; and (b) if p is a periodic point off then dim Ws(p;f) = dim Ws(h(p); g) Property (b) is not usually included in the definition of Q-stable (see (3)), but it is a weak condition which is very natural and is apparently necessary for the proof of one of our lemmas (22) In his paper (4), S Smale provides sufficient conditions for a diffeomorphism to be £2-stable One of his conditions is that the nonwandering set have a hyperbolic structure Recall that a closed invariant set A is said to have a hyperbolic structure if (a) There is continuous splitting of the restriction of the tangent bundle to A,