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
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
The micro-local lift -- finding a measure invariant under the
geodesic flow A.
We will discuss a construction of the micro-local lift. For
quantum limit on the quotient Gamma \ H of the hyperbolic plane H
lattice Gamma one can find an A-invariant measure on Gamma \
(which pushed down to Gamma \ H coincides with the micro-local
This will roughly follow the IMRN-article (2001, No.17) by
Lindenstrauss, see number 12. on
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
properties of the micro-local lift.
As there are many A-invariant measures on Gamma \ SL_2(R)
have to discuss recurrence and establish it for the micro-local
of an arithmetic quantum limit (defined by joint eigenfunctions of
Laplacian and the Hecke-operators). This is contained in the
Annals, 163--2006 by Lindenstrauss, which is number 20. on the
website, and as we will see already eliminates the possibility of
certain A-invariant measures in the micro-local lift. For this we
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
As one currently doesn't know whether the above two
enough to characterize the Haar measure on Gamma \ SL_2(R) we will
also discuss entropy. In the paper Comm. Math. Phys. 233 (2003),
1, number 13. on the above website, Bourgain and Lindenstrauss show
positive entropy of all ergodic components. In the Annals paper
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
\ H. As time permits we will review entropy theory for ergodic
theoretic systems, discuss how to establish positive entropy, and
As an additional source let me mention an ergodic theory book in
preparation, which is available online at
For the most part of the above discussion we will not need the
theory presented there, but Chapter 9 may be useful as it discusses
the geodesic flow on Gamma \ SL_2(R).