Bootstrap for Local Rigidity of Anosov Automorphisms on the 3-Torus

Abstract

We establish a strong form of local rigidity for hyperbolic automorphisms of the 3-torus with real spectrum. Namely, let \({L:\mathbb{T}^3\to\mathbb{T}^3}\) be a hyperbolic automorphism of the 3-torus with real spectrum and let f be a C 1 small perturbation of L. Then f is smoothly (\({C^\infty}\)) conjugate to L if and only if obstructions to C 1 conjugacy given by the eigenvalues at periodic points of f vanish. By combining our result and a local rigidity result of Kalinin and Sadovskaya (Ergod Theory Dyn Syst 29:117–136, 2009) for conformal automorphisms this completes the local rigidity program for hyperbolic automorphisms in dimension 3. Our work extends de la Llave–Marco–Moriyon 2-dimensional local rigidity theory (Commun Math Phys 109:368–378, 1987; Ergod Theory Dyn Syst 17(3):649–662, 1997; Commun Math Phys 109(4):681–689, 1987).