Mathematisches Institut
der LMU, München

Prof. D. Kotschick, D. Phil.

Prof. Dr. T. Vogel

Seminar on Anosov systems - Sommersemester 2018

Geometry of Anosov Systems

A diffeomorphism f of a manifold is said to be Anosov if there is a continuous splitting of the tangent bundle into two complementary invariant subbundles such that the differential of f is uniformly expanding on one of the summands and uniformly contracting on the other one. Taking the suspension of such an f we obtain a flow on its mapping torus for which a splitting as above exists for the quotient of the tangent bundle by the one-dimensional distribution spanned by the flow. This is the simplest example of an Anosov flow. Other classical examples include the geodesic flows on the unit tangent bundles of negatively curved Riemannian manifolds.

Anosov systems are a very rich class of dynamical systems with remarkable dynamical and geometric properties. On the dynamical side, Anosov systems illustrate concepts like hyperbolicity, structural stability, and various ergodic properties. On the geometric side, there are interesting relations not just with Riemannian geometry, but also with the theory of foliations and with contact and symplectic geometry. These will be the main topic of the seminar. We shall start with basic definitions and examples, continue with some foundational results concerning the integrability of the expanding and contracting distributions and structural stability, and then explore the differential geometry of Anosov systems.

Prospective participants should contact one of the organizers, preferably by email. There will be an organizational meeting on Wednesday, April 11^th, at 10:15 AM (=10 Uhr c.t.), in room B 046. If you are interested in this seminar, then please come to the organizational meeting, even if the time is not suitable for you, so that we can discuss the timing, and perhaps choose a better time slot for subsequent weeks.

Plan of talks


General References:
[A] D. V. Anosov: Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). American Mathematical Society Translations, Providence, R.I. 1969.
[AA] V. I. Arnold and A. Avez: Ergodic problems of classical mechanics. W. A. Benjamin, Inc., New York-Amsterdam 1968.
[KH] A. Katok and B. Hasselblatt: Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995.

There will be further, more specific, references for some of the talks.
D. Kotschick and T. Vogel