Uniformization and an Index Theorem for 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 an isometric diffeomorphism. The key to the solution is the method of uniformization: We assign to the nonlocal problem a pseudodifferential operator with the same index, acting in 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. 
Anton Savin, Elmar Schrohe, Boris Sternin
math.AP math.KT math.OA