Hyperbolicity in geometry and homogeneous dynamics

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Next lectures
09.12.2025, 15:15 (S311)
16.12.2025, 15:15 (S311)
23.12.2025, 15:15 (S311)
30.12.2025, 15:15 (S311)
06.01.2026, 15:15 (S311)
13.01.2026, 15:15 (S311)
20.01.2026, 15:15 (S311)
27.01.2026, 15:15 (S311)
03.02.2026, 15:15 (S311)

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We are interested in fine structural and statistical features of certain continuous dynamical systems of a geometric origin. Our motivating examples are the geodesic and horocyclic flows on the unit tangent bundle of a surface with a complete, negatively curved metric. The simplest case to consider is when the surface is compact or finite volume, and the metric has curvature everywhere equal to -1, i.e., the surface is hyperbolic.

We will prove that the geodesic flow on such a closed hyperbolic surface is exponentially mixing, has positive entropy, is uniformly hyperbolic (in fact Anosov), and thus exhibits both structural stability and chaos. It can also be coded using a Markov partition and studied in terms of symbolic dynamics, and we will use this coding to compute the growth rate of periodic orbits of the flow, among other applications.

The geodesic flow on a complete hyperbolic surface is an example from the world of homogeneous dynamics. The basic objects are: a (non-compact) Lie group G, a discrete subgroup Γ ≤ G, and a 1-parameter subgroup A ≤ G acting continuously on Γ \G preserving a finite invariant (Haar) measure. Sometimes the dynamics of this action can be used to count orbits of Γ on G/A. We will explore some examples of homogeneous dynamical systems and counting problems. Some of the motivation comes from number theory. Homogeneous dynamics is a very active field of current research - especially the dynamics of unipotent flow (like the horocycle flow). This course will serve as an introduction to some of the basic principles with an emphasis on (hyperbolic) geometry.