Instructor: James Marshall Reber.
Syllabus: Review of topological dynamical systems (discrete and continuous) with examples. Definition of Anosov dynamical systems with examples. Cone-criterion for hyperbolicity. Further examples of Anosov dynamical systems (geodesic flows on negatively curved Riemannian manifolds, hyperbolic toral automorphisms, magnetic flows, etc.). Properties of Anosov systems (shadowing, expansivity, closing, specification, Axiom A). Livsic theorem (examples of applications).
Schedule: We will have three lectures per week, on Mondays, Wednesdays, and Fridays, from 11 a.m. to noon, starting on June 27th, and ending on July 15th. Online on Zoom.
Recordings:
- Week 1: lecture 1, lecture 2, lecture 3;
- Week 2: lecture 4, lecture 5, lecture 6;
- Week 3: lecture 7, lectures 8 and 9 (combined).
Main references:
- Lecture notes written by James specifically for this course.
- Brin, M.; Stuck, G.; Introduction to dynamical systems. Corrected paper back edition of the 2002 original. Cambridge University Press, Cambridge, 2015. xii+247 pp. ISBN: 978-1-107-53894-8; 978-0-521-80841-5.
Extra references:
- Katok, A.; Hasselblatt, B.; Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995. xviii+802 pp. ISBN: 0-521-34187-6.
- Fisher, T.; Hasselblatt, B.; Hyperbolic Flows. Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2019. 737 pp. ISBN: 978-3-03719-200-9.
- Barreira, L; Valls, C.; Dynamical systems: An introduction. Translated from the 2012 Portuguese original. Universitext. Springer, London, 2013. x+209 pp. ISBN: 978-1-4471-4834-0; 978-1-4471-4835-7.