Global smooth rigidity for toral automorphisms

We study regularity of a conjugacy between a hyperbolic or partially hyperbolic toral automorphism $L$ and a $C^{\mathrm{\infty }}$ diffeomorphism $f$ of the torus. For a very weakly irreducible hyperbolic automorphism $L$ we show that any $C^1$ conjugacy is $C^{\mathrm{\infty }}$. For a very weakly irreducible ergodic partially hyperbolic automorphism $L$ we show that any $C^{1+\text{H}\ddot{\text{o}}\text{lder}}$ conjugacy is $C^{\mathrm{\infty }}$. As a corollary, we improve regularity of the conjugacy to $C^{\mathrm{\infty }}$ in prior local and global rigidity results.