4 problems

Let $X_0=(1,b1)$ be a constant vector field on $a4^2$, and let $(T_t)$ be the flow generated by the reparametrized vector field $X=c6X_0$, where $c6:a4^2\to R_{\geq 0}$ is a $C^3$…

Katok–Herman conjecture. K.A.M. theory is perturbation theory for rotations in tori, and only rotations in tori can serve as its linear model.

Katok–Herman conjecture. The only examples, up to smooth diffeomorphism and conjugacy, of $CR(M)$ diffeomorphisms are Diophantine translations in tori.

Torus-rotation coding conjecture. In these slowly growing cases, the Arnoux–Rauzy system is conjectured to be a coding of a rotation of the $2$-torus.