Homogeneous dynamics on surfaces

Practical information:

We will meet Tuesday and Thursday from 9-11 in INF 205 SR 9 starting June 14 and ending July 19, 2022.

The exam will take place (by appointment) during the weeks of July 25 and August 2.

Information about credits can be found here and here.

Prerequisites:

I will assume knowledge from a first course in complex analysis, algebraic topology (namely covering space theory), basic differential geometry and smoth manifolds, measure theory. I will also assume familiarity with groups and group actions. Background in hyperbolic geometry would be useful but is not required.

Description and plan:

We will examine the geometry and dynamics of the geodesic and horocycle flows on the unit tangent bundle of finite volume hyperbolic surfaces, i.e., two dimensional manifolds equipped with a Riemannian metric of constant negative curvature. We plan to prove that the geodesic flow is ergodic (chaotic in the measure theoretic sense) and that every orbit of the horocycle flow is either cusp parallel or dense. As an application, we discuss a relationship between the geodesic flow on the modular surface and continued fraction expansions of irrational numbers.

The plan of the course is to begin by discussing the Poincaré model for hyperbolic geometry, understanding the isometries as well as the geometry of the geodesic and horocyle flows. Then we discuss finitely generated discrete (i.e. Fuchsian) subgroups; the quotient of the plane by such a discrete group gives rise to a hyperbolic surface. Reveiwing concepts from ergodic theory as needed, we then show that the geodeisc and horocycle flows are chaotic, under suitable assumptions. The remainder of the course is decidated to applications.