We introduce the notion of abstract angle at a couple of points defined by two radial foliations of the closed annulus. We will use for this purpose the digital line topology on the set \({\mathbb{Z}}\) of relative integers, also called the Khalimsky topology. We use this notion to give unified proofs of some classical results on area preserving positive twist maps of the annulus by using the Lifting Theorem and the Intermediate Value Theorem. More precisely, we will interpretate Birkhoff theory about annular invariant open sets in this formalism. Then we give a proof of Mather’s theorem stating the existence of crossing orbits in a Birkhoff region of instability. Finally we will give a proof of Poincaré – Birkhoff theorem in a particular case, that includes the case where the map is a composition of positive twist maps.

我们在由封闭环的两个径向叶理定义的几个点处引入了抽象角度的概念。为此，我们将使用集合\({\mathbb{Z}}\)上的数字线拓扑相对整数，也称为 Khalimsky 拓扑。我们利用这个概念，利用提升定理和中值定理，对环面保正扭曲图的一些经典结果进行了统一证明。更准确地说，我们将在这种形式主义中解释关于环形不变开集的伯克霍夫理论。然后我们给出马瑟定理的证明，证明在 Birkhoff 不稳定区域中存在交叉轨道。最后，我们将在特定情况下给出 Poincaré – Birkhoff 定理的证明，其中包括映射是正扭曲映射的组合的情况。