Abstract
We consider a hyperbolic toral automorphism L and its C 1 -small perturbation f. It is well-known that f is Anosov and topologically conjugate to L, but a conjugacy H is only Hölder continuous in general. We discuss conditions for smoothness of H, such as conjugacy of the periodic data of f and L, coincidence of their Lyapunov exponents, and weaker regularity of H, and we summarize questions, results, and techniques in this area. Then we introduce our new results: if H is weakly differentiable then it is C 1+Hölder , and if L is also weakly irreducible then H is C ∞ .
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