transactions of the american mathematical society Volume 158, Number 2, August 1971 NECESSARY CONDITIONS FOR STABILITY OF DIFFEOMORPHISMS BY JOHN FRANKSO) 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 Arthroots 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 (2.2). 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, TMA= ES © Eu which is preserved by the derivative dfi. (b) There exist constants C>0, C>0 and Ae(0, 1) and a Riemannian metric || || on TMK such that \\dfn(v)\\ < C\n\\v\\ for v e Es and « > 0, and \\dfn(v)\\ £ C'A-nH| for v e Eu and « > 0. One would like to prove that the condition above is necessary for Q-stability. In this paper we give results which are a start in that direction. I would like to thank Presented to the Society, January 24, 1971; received by the editors February 18, 1970 and, in revised form, October 27, 1970. AMS 1970 subject classifications. Primary 58F10, 58F15. Key words and phrases, fí-stability, structural stability, hyperbolic structure, non- wandering set. C) Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research under Contract No. F44620-67-C-0029. Copyright © 1971, American Mathematical Society 301