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).

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).