  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 H\"older 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 $C^{1+\text{H\"older}}$ and, if $L$ is also weakly irreducible, then $H$
is $C^\infty$.
