Elliptic theory for operators associated with diffeomorphisms of smooth manifolds

In this paper, the authors give a survey of index theory for elliptic operators associated with diffeomorphisms of smooth manifolds. Mostly, they consider operators associated with a discrete group G of diffeomorphisms of a closed smooth manifold M, called G-pseudodifferential operators. These operators act on the space C^infty(M) of smooth functions on M and have the form D=sum_{gin G}D_gT_g, where {D_g} is a collection of pseudodifferential operators on M and T_gu(x)=u(g^{-1}(x)) is the shift operator corresponding to the diffeomorphism gin G. The authors introduce the notion of ellipticity for G-pseudodifferential operators, show that ellipticity implies the Fredholm property of the operator in Sobolev spaces, and prove the index formula, which computes the index of an elliptic G-pseudodifferential operator in terms of the symbol of the operator and the topological invariants of the G-manifold M. Both isometric and nonisometric actions are treated. The authors also discuss operators associated with an action of a compact Lie group G on a closed smooth manifold M. These operators have the form D=int_GD_gT_gdg, where dg is the Haar measure. The authors describe a method of pseudodifferential uniformization which allows them to reduce the index problem for elliptic G-pseudodifferential operators to the index problem for transversally elliptic operators acting on sections of some infinite-dimensional bundles over M.