**Course Description:**

The aim of this course is to develop, from scratch, classical and quantum dynamics in the setting of hyperbolic surfaces. Flows on hyperbolic surfaces can be introduced with minimal prerequsites, but at the same time they exhibit surprisingly rich chaotic properties such as ergodicity and mixing. From the quantum point of view these chaotic properties are reflected in the asymptotic behaviour of high-energy eigenstates of the Laplace operator which are of fundamental importance in mathematical physics and the theory of automorphic forms.

In this course we only assume basic knowledge of analysis and start with an elementary discussion of
hyperbolic geometry and constructions of hyperbolic surfaces.
Then we introduce the geodesic and horocycle flows and investigate distribution of their orbits.
In the second part of the course we turn to discussion of quantum phenomena.
This involves the study of eigenfunctions φ_{λ} for the Laplace operator Δ.
It turns out that the chaotic properties of the geodesic flow are reflected in the
asymtotic behaviour of the eigenfunctions φ_{λ} as λ→∞.
We explain, in details, such phenomena as the quantum ergodicity and the quantum
unique ergodicity in the setting of arithmetic surfaces.

The highlight of the course would be an outline Lindenstrauss' proof of arithmetic quantum unique ergodicity.

We plan to cover the following topics:

- basic hyperbolic geometry,
- geometric and arithmetic examples of hyperbolic surfaces,
- geodesic and horocycle flows,
- ergodicity and mixing properties,
- invariant measures for the horocycle flow,
- Casimir and Laplace operator,
- microlocal lift and quantum ergodicity,
- Hecke operators and arithmetic quantum unique ergodicity.

** Time: ** Friday 10-12am; See TCC Timetable

** Lecture notes: **

- Lecture 1-2: Hyperbolic plane and hyperbolic surfaces
- Lecture 3: Geodesic flow: mixing, Anosov closing lemma
- Lecture 4: Horocycle flow: mixing, unique ergodicity
- Lecture 5: Laplace operator and its spectral decomposition
- Lecture 6: Trace formula and the Weyl law
- Lecture 7: Microlocal lift and quantum ergodicity
- Lecture 8: Hecke operatators and recurrence of eigenstates
- Lecture 9: Entropy of quantum limits

** Homework Problems: **

** Course Assessment: ** The course is assessed by the above problem sheets.
Solutions for 10 problems have to be submitted by April 22.

**References:**

- B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces. Cambridge University Press, 2000.
- P. Buser, Geometry and spectra of compact Riemann surfaces. Birkhauser, 1992.
- F. Dal'Bo, Geodesic and horocyclic trajectories. Universitext, 2011.
- M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces. Clay Math. Proc. 10, pp. 155-241, 2010.
- M. Einsiedler and T. Ward, Arithmetic quantum unique ergodicity. Lecture notes, 2010.
- M. Einsiedler and T. Ward, Ergodic theory with a view towards number theory. Springer, 2011.
- Introduction to Hyperbolic Surfaces. Lecture notes, 2012.
- P. Sarnak, Arithmetic quantum chaos. Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995.
- P. Sarnak, Spectra of hyperbolic surfaces. Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 4, 441-478.