Uniformization and index of elliptic operators associated with diffeomorphisms of a manifold

We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely, differential operators with shifts induced by the action of a (not necessarily periodic) isometric diffeomorphism. The key to the solution is the method of uniformization. To the nonlocal problem we assign a pseudodifferential operator, with the same index, acting on the sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah—Singer index theorem. © 2015, Pleiades Publishing, Ltd.