Constructing Structurally Stable Diffeomorphisms

Abstract

One of the central problems of geometric dynamics is to understand the relation of the dynamics of a system to the topology of the manifold on which it occurs-what kinds of dynamics can occur on which manifolds. We are concerned here with this problem as it relates to Smale diffeomorphisms (structurally stable, Axiom A diffeomorphisms with zero dimensional basic sets, see ? 1 for all definitions). This is a large class and includes a representative of every isotopy class. For these diffeomorphisms, the action on a basic set (which is the dynamics in which we are interested) is topologically conjugate to a subshift of finite type and hence specified entirely by a non-negative integer matrix. Our question then becomes: On a given compact manifold M when does there exist in a given homotopy class a diffeomorphism f whose basic sets are the subshifts of finite type corresponding to a set of pre-assigned matrices and occurring with pre-assigned index? (See [10] also.) For the homotopy class of the identity on a large class of manifolds (including all simply-connected manifolds of dimension greater than five with torsion-free homology) we are able to answer this question up to a finite power. That is, we give necessary and sufficient conditions for the existence of a diffeomorphism f isotopic to the identity whose restrictions to its basic sets are the kth powers of certain pre-assigned subshifts of finite type, each occurring with pre-assigned index. These necessary and sufficient conditions are immediately checkable and depend only on the matrices given and the Betti numbers of the manifold. A somewhat more general sufficient condition for the existence f can be given in terms of the existence of a Morse function on M with certain type numbers. Recall that the jt type number of a Morse function is the number of critical points of index j it has. Since the direct sum of matrices corresponds to the disjoint union of the corresponding subshifts of finite type, we can describe all basic sets of a given index (see definition in ? 1) for a Smale diffeomorphism in a single matrix.