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.

Prerequisites. 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.

Part 1: Hyperbolic geometry 
               (1) Hyperbolic plane, the Poincaré upper half plane model H2, the boundary at infinity.
               (2) Isometries, geodesics and horospheres.
               (3) The PSL2(R) interpretation.

Part 2: Hyperbolic surfaces
               (1) Basic definitions; hyperbolic surfaces as a quotient of H2 by a group Γ.
               (2) The cross ratio and existence of hexagons with right angles.
               (3) Existence of hyperbolic surfaces. Classification of hyperbolic surfaces. 

Part 3: Dynamics

(1) The action of Γ on the boundary at infinity. on the boundary at infinity.

(2) The geodesic flow: The Anosov property and the closing lemma.

(3) The horospherical flow.

Part 4: Measure theoretic approach

(1) Invariant measures and currents.

(2) The Liouville measure and the Poincaré recurrence theorem: topological transitivity.

(3) Ergodicity and mixing; the spectral approach: Moore Theorem.

(4) Unique ergodicity of the horospherical flow.

Part 5: Counting geodesics

(1) Equidistribution and mixing

(2) Counting geodesics and the entropy