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

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