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 $C^r$ diffeomorphims can be conjugated to the affine action by $C^{r-ε}$ conjugacy. Next, we show that in any dimension, $C^1$ small perturbations can be conjugated to an affine action via $C^{1+ε}$ 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.
设$BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$是$n \geq 2$的可解Baumslag-Solitar群。我们研究了$BS(1, n)$在平面上的保向同胚表示，假设$a$是线性映射，$b$是有界位移映射.我们发现，忠实行动的可能性在很大程度上取决于若尔当典范形的地图$h$定义的行动$a$。如果$h$在$\mathbb R$上可对角化，我们将给出例子或证明依赖于特征值的刚性定理。我们还表明，在$h$是椭圆或抛物线的情况下，一些刚性。然后利用环面的同胚给出了$BS(1, n)$作用的应用.
