PERHAPS THE simplest smooth discrete time dynamical systems are the Morse-Smale diffeomorphisms. Among structurally stable systems they exhibit the simplest recurrent behavior-a finite set of hyperbolic periodic pionts and no other recurrence. These systems have been the object of considerable research. Palis and Smale[l3] proved that they are structurally stable. More recent work has dealt with the relationship of the homotopy class of a diffeomorphism to the kind of dynamics it exhibts [5,9, 11, 161 and the question of the existence of a Morse-Smale diffeomorphism in a given homotopy class. This latter topic is the subject of this article. In[l6], Shub and Sullivan showed among other things that a necessary condition for the existence of a Morse-Smale diffeomorphism is that all eigenvalues of the induced maps on homology be roots of unity. In the case of simply connected manifolds of dimension greater than five. they reduced the question of existence to an algebraic condition on the chain level for the diffeomorphism (1.2, below) and using this condition pionted out the existence of additional obstructions related to the ideal class groups of the cyclotomic fields. In this article we identify the group in which the obstruction lies, express it in terms of the algebraic K-theory of the induced endomorphisms f*: H,(M)+H,(M) and show that there are no further obstructions. More precisely, we consider the category QI of abelian groups with quasi-idempotent endomorphisms (i.e. having all eigenvalues zero or roots of unity) and show the obstruction lies in the torsion subgroup G of Ko(QI). The obstruction depends in fact only on the elements d(Lf*k]) in G, where Lf*k] denotes the class in Ko(QI) of the quasi-idempotent endomorphism f*L: H,(M, Z) + H,(M, Z) and c$: K,(QI)+ G is a projection onto the summand G. Of course if f is a diffeomorphism f *Ir will be quasi-unipotent (having only roots of unity as eigenvalues) since it is an automorphism. Our main result is the following.