Finite periodic data rigidity for two-dimensional area-preserving Anosov diffeomorphisms

Let f , g f,g be C 2 C^2 area-preserving Anosov diffeomorphisms on T 2 \\mathbb {T}^2 which are topologically conjugate by a homeomorphism h h ( h f = g h hf=gh ). We assume that the Jacobian periodic data of f f and g g are matched by h h for all points of some large period N ∈ N N\\in \\mathbb {N} . We show that f f and g g are “approximately smoothly conjugate.” That is, there exists a C 1 + α C^{1+\\alpha } diffeomorphism h ¯ N \\overline {h}_N such that h h and h ¯ N \\overline {h}_N are