Rigidity of some abelian-by-cyclic solvable group actions on \({\mathbb{T}}^N\) (Q2184743)

The authors study a natural class of groups that act as affine transformations of the torus \(\mathbb{T}^N\). They investigate whether these groups can act smoothly and nonaffinely on \(\mathbb{T}^N\) while remaining homotopic to the affine actions. In the case of affine actions, elliptic and hyperbolic dynamics coexist, forcing complicated dynamics in nonaffine perturbations. Using KAM theory, the authors show that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine conditions on the action are fulfilled. Under natural dynamical hypotheses, they get a complete classification of such actions in dimension two. Specifically, any such group action by \(C^r\)-diffeomorphisms can be conjugated to the affine action by \(C^{r-\varepsilon}\)-conjugacy. Finally, the authors show that in any dimension, \(C^1\)-small perturbations can be conjugated to an affine action via \(C^{1-\varepsilon}\)-conjugacy.