Smooth rigidity for very non-algebraic Anosov diffeomorphisms of codimension one

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 C¹ 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.