  In this paper we introduce a new methodology for smooth rigidity of Anosov
diffeomorphisms based on "matching functions." The main observation is that
under certain bunching assumptions on the diffeomorphism the periodic cycle
functionals can provide such matching functions. For example we consider a
sufficiently small C^1 neighborhood of a linear hyperbolic automorphism of the
3-dimensional torus which has a pair of complex conjugate eigenvalues. Then we
show that two very non-algebraic (an open and dense condition) Anosov
diffeomorphisms from this neighborhood are smoothly conjugate if and only they
have matching Jacobian periodic data. We also obtain a similar result for
certain higher dimensional codimension one Anosov diffeomorphisms.
