The course will start with an elementary introduction to hyperbolic geometry: geodesics, isometries, horospheres. We will emphasize the boundary at infinity and the cross ratio. Then we will move to the construction and classification of hyperbolic surfaces, through the construction of right-angled hexagons.
We then introduce the main dynamical players of our course: the geodesic flow of a closed surface and the action of its fundamental group on the boundary at infinity. We will first explain the Anosov property and its consequence: the closing lemma. The measure theoretic properties of the geodesic flow play a fundamental role in this course. We introduce invariant measures and currents and explain ergodicity and mixing using a spectral approach through Moore Theorem.
In the last part, we will explain, in this elementary case, Margulis' fundamental ideas and use it to prove the equidistribution of closed geodesics and compute the entropy. A familiarity with the fundamental group and coverings would be helpful although not strictly necessary. No Riemannian geometry will be involved (notably the definition of curvature and its computations) since our approach will be metric only. Nevertheless a preliminary knowledge of Riemannian geometry could help to put the course in perspective.
The course is based on the following incomplete set of notes which will be expanded and clarified during the lectures.
