Inventiones math. 48, 279-292 (1978) I~ventiones mathematicae ~) by Springer-Veriag 1978 The Periodic Behavior of Morse-Smale Diffeomorphisms John Franks* and Carolyn Narasimhan Department of Mathematics, Northwestern University, Evanston, Illinois 60201, USA Morse-Smale diffeomorphisms are the simplest examples of structurally stable dynamical systems and exhibit the simplest recurrent behavior-a finite set of periodic orbits. They have been much studied; Palis and Smale [PS] for example showed that they are structurally stable and in fact precisely the set of structurally stable diffeomorphisms with finitely many periodic points. In [SS], Shub and Sullivan investigated (among other things) the question of which isotopy classes admit Morse-Smale diffeomorphisms and what the possibilities were for the periodic behavior of these diffeomorphisms. They reduced these problems to algebraic conditions on endomorphisms of chain complexes ((2.1) below) which form the starting point of this article. We are concerned with the question of the existence of Morse-Smale diffeomorphisms in a given homotopy class with prescribed periodic data-that is with periodic orbits of specified period and index (dimension of unstable manifold). In [F] the first author gave necessary conditions ((1.3) below) for this existence in terms of the induced maps J.k: Hk(M)~Hk(M) which are quite elementary and easily checkable. In IN] the second author showed that in the homotopy class of the identity on two-dimensional manifolds these conditions are sufficient as well as necessary. In this article we prove, in Theorem (2.4) below, this same result for a large number of homotopy classes (including that of the identity) on all simply connected manifolds of dimension greater than five with torsion free homology, if there are no periodic orbits of index 1 or (n-1). Thus for many manifolds including spheres of dimension greater than five, we have a nearly complete characterization of the kinds of periodic behavior which can occur in a Morse- Smale diffeomorphism. w 1. Periodic Data and Necessary Conditions We begin by recalling a few definitions. If f: M-~ M is a diffeomorphism and x6M then x is said to be chain-recurrent provided that given any ~>0 there * Research supported in part by NSF Grant MCS77-01080 0020-9910/78/0048/0279/$02.80