Twist Maps of the Annulus: An Abstract Point of View

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.