Index problem for elliptic operators associated with a diffeomorphism of a manifold and uniformization

Index problem for elliptic operators associated with a diffeomorphism of a manifold and uniformization was studied. Let M be a smooth manifold on which a smooth isometric diffeomorphism g: M → M is given. The powers of this diffeomorphism generate an action of the group &Zdbl; on the manifold. The method for solving this problem in pseudodifferentially uniformizing the operator D, reducing this operator to some elliptic differential operator whose index coincides with the index of the original operator. The reduction is performed in two stages. The first stage consists in replacing the manifold M by the manifold M × &Rdbl; with the diagonal action of the group &Zdbl;. At the second stage, the obtained operator on M × &Rdbl; as a differential operator on the sections of an infinite-dimensional bundle on the smooth manifold.