Abstract : The aim of this course is to survey many striking developments in recent years related to horocyclic orbits. In this course we appreciate some of these recent advances. For example, we will see a result of Alexander Bellis in his thesis where he shows that the horocyclic orbit could be contained strictly in its corresponding strongly stable manifold. We will talk also a recent result of Farre, Landesberg and Minsky. They discovered a quite surprising result, which states that slight changes to the geometry could dramatically change the topology of no-dense horocyclic orbit closures. We hope at the end of this course, the students and young researchers will be strongly equipped to make new original contributions and see how to study horocyclic flow on flat surfaces and surfaces with variable curvature. The course content is as follows :

Session 1 : Horocyclic flow : Different view points In the first session background will be installed. Geometry and topology of hyperbolic surfaces of infinite type completed by exercises for the student and open question about curves on the surfaces. We also evoke the classification of limit points of the fundamental groups of the surfaces without to recall what is done in the case of finite type.

Session 2 : Topology of horocyclic orbits In this session after evoking the different possibilities for the horocyclic orbits we establish a dictionary between the topological properties of the horocyclic orbits and their corresponding limit points. After that we will study the structure of horocyclic orbits closure and end with the question of minimal sets for the horocyclic flow .

Session 3 : Relation between w(u) and h(u) Until recently, the horocyclic orbit was confounded to the strongly stable manifold of the geodesic flow. Alexandre Bellis, using the injectivity radius along a geodesic ray, gives an example where the horocyclic orbit and the strong stable manifold are distinct. In this session, we'll try to understand this example and draw a conjecture about the equality of the two objects.

Session 4 : Rigidity of horocyclic orbit closure In the session the hyperbolic structure on an infinite type topological surface is fixed. Here we show that some rather surprising phenomena occur when we slightly disturb the hyperbolic structure.

Session 5 : Ergodicity of the horocyclic flow The last session discuses the ergodic for the horocycle flow of infinite type hyperbolic surfaces. It is based in one hand on Omri Sarig work's and on a work of Lindestrauss and Landesberg.