Quantum chaos on arithmetic hyperbolic surfaces : Lindenstrauss' result (Manfred Einsiedler, Ohio State University)

We will discuss the proof of Arithmetic Quantum Unique Ergodicity for
certain arithmetic quotients of the hyperbolic plane given by E.
Lindenstrauss. His proof uses ergodic theoretic methods, but for the
most part we will focus on the geometric and number theoretic
arguments that establish the needed ingredients for applying the
ergodic theoretic theorem. Here is the list of the material that will
be discussed.

The micro-local lift -- finding a measure invariant under the geodesic flow A.
We will discuss a construction of the micro-local lift. For any
quantum limit on the quotient Gamma \ H of the hyperbolic plane H by a
lattice Gamma one can find an A-invariant measure on Gamma \ SL_2(R)
(which pushed down to Gamma \ H coincides with the micro-local lift).
This will roughly follow the IMRN-article (2001, No.17) by
Lindenstrauss, see number 12. on
http://www.math.princeton.edu/~elonl/Publications/index.html . In
order to follow that article we will have to review H, SL_2(R), and
the universal enveloping algebra of sl_2(R).

Recurrence under the Hecke-tree -- establishing the first additional
properties of the micro-local lift.
As there are many A-invariant measures on Gamma \ SL_2(R) we will
have to discuss recurrence and establish it for the micro-local lift
of an arithmetic quantum limit (defined by joint eigenfunctions of the
Laplacian and the Hecke-operators).  This is contained in the paper
Annals, 163--2006 by Lindenstrauss, which is number 20. on the above
website, and as we will see already eliminates the possibility of
certain A-invariant measures in the micro-local lift. For this we will
study the relationship between PGL_2(Q_p), (p+1)-regular trees, the
Hecke-operators, and its eigenfunctions.

Positive entropy -- the second additional property. Applying a measure
classification.
As one currently doesn't know whether the above two properties are
enough to characterize the Haar measure on Gamma \ SL_2(R) we will
also discuss entropy. In the paper Comm. Math. Phys. 233 (2003), no.
1, number 13. on the above website, Bourgain and Lindenstrauss show
positive entropy of all ergodic components. In the Annals paper (20.)
Lindenstrauss uses all of these to prove a measure classification
which establishes that the micro-local lift is the Haar measure and
hence that the arithmetic quantum limit is the volume measure on Gamma
\ H. As time permits we will review entropy theory for ergodic
theoretic systems, discuss how to establish positive entropy, and the
measure classification.

As an additional source let me mention an ergodic theory book in
preparation, which is available online at
http://www.mth.uea.ac.uk/ergodic/
For the most part of the above discussion we will not need the ergodic
theory presented there, but Chapter 9 may be useful as it discusses
the geodesic flow on Gamma \ SL_2(R).