Rigidity of Some Abelian-by-Cyclic Solvable Group Actions on $${\mathbb {T}}^N$$

Abstract

In this paper, we study a natural class of groups that act as affine transformations of $${\mathbb {T}}^N$$ . We investigate whether these solvable, “abelian-by-cyclic," groups can act smoothly and nonaffinely on $${\mathbb {T}}^N$$ while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, 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 met. In dimension two, under natural dynamical hypotheses, we get a complete classification of such actions; namely, any such group action by $$C^r$$ diffeomorphisms can be conjugated to the affine action by $$C^{r-\varepsilon }$$ conjugacy. Next, we show that in any dimension, $$C^1$$ small perturbations can be conjugated to an affine action via $$C^{1+\varepsilon }$$ conjugacy. The method is a generalization of the Herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics.