Let $T_a:x\mapsto x+a$ be a Diophantine translation on the torus $\mathbb T^d$, $d>1$. Let $h$ be some $C^1$ diffeomorphism of $\mathbb T^d$ such that $$ g=h\circ T_a\circ h^{-1} $$ is $C^\infty$. Is it true that $h$ is $C^\infty$? Was this question discussed in the literature?

Comments

- For $d=1$ this is a corollary of a more general theorem of Herman.
- In his "Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations" Herman also discussed the toral case, but it doesn't seem that my question was addressed there. But he has $g\in C^\omega, h\in C^\infty$ implies $h\in C^\omega$.