  We develop elliptic theory of operators associated with a diffeomorphism of a
closed smooth manifold. The aim of the present paper is to obtain an index
formula for such operators in terms of topological invariants of the manifold
and of the symbol of the operator. The symbol in this situation is an element
of a certain crossed product. We express the index as the pairing of the class
in K-theory defined by the symbol and the Todd class in periodic cyclic
cohomology of the crossed product.
