Lectures of Anatole Katok

ALGEBRAIC ACTIONS
OF
HIGHER RANK ABELIAN GROUPS
AND
INTRODUCTION TO RIGIDITY

Lectures of Anatole Katok

26 February - 8 March 2007
Banach Center, Warszawa, Poland

ANATOLE KATOK will read a minicourse at Banach Center in Warsaw. 6 lectures, 1,5 hours each, are planned

Mondays - Feb. 26, March 5, 12:15, room 403,
Wednesdays - Feb. 28, March 7, 12:15, room 403,
Thursdays - March 1, 8, 12:15, room 403

and, depending on a need, some tutorials/exercises.

We invite mainly Phd/MSc students. We offer a number (limited, available according to the order of applications) of free accommodation places in the Banach Center guest rooms and, if there is a need, a cheap accommodation elsewhere. There will be no registration fee. We enclose an application form at the end of the advertisement.

The first goal of this course is to describe the objects of extensive recent and on-going research in rigidity theory for group actions, including necessary background from Lie theory and algebraic number theory. After that we prove model results in differentiable rigidity and measure rigidity which illustrate some of the methods used in this area. Along the way we will introduce necessary background from ergodic theory and hyperbolic dynamics.

The course will be accessible to students with a solid background in real analysis, advanced linear algebra and basic geometry/topology, including elementary properties of differentiable manifold. Basic acquaintance with ergodic theory will be helpful but not strictly necessary. No previous knowledge of Lie groups, algebraic number theory of hyperbolic dynamics is required.

Chapters from forthcoming book

DIFFERENTIABLE RIGIDITY OF HIGHER RANK ABELIAN GROUP ACTIONS joint with Viorel Nitica (to appear in Cambridge University Press) as well as from some published articles will be provided as course notes.


            SYLLABUS

1. PRELIMINARIES (approx. one lecture)

1.1. Differentiable,  topological and measure-preserving group actions.
     Functorial constructions: restriction, Cartesian product, factor,
     suspension, natural extension, skew product.

1.2. Linear actions of higher rank abelian groups. Roots, Lyapunov
     exponents and Weyl chambers, hyperbolic and partially hyperbolic actions.

1.3. Elements of Lie groups.  Lie algebra, exponential map. Examples of
     linear Lie groups.  Definitions and examples of lattices in Lie groups


2. PRINCIPAL CLASSES OF ALGEBRAIC ACTIONS (approx. three lectures)

2.1. Automorphisms, homogeneous and affine actions. Definitions and first
     examples

2.2. Automorphism of the torus. Equivalent forms of ergodicity and
     partial hyperbolicity conditions.  Harmonic analysis method for studying
     dynamical and ergodic properties of automorphisms of the torus.

2.3  Algebraic centralizer  of a toral automorphism. Connection with units
     in algebraic number fields. Dirichlet unit theorem. Linear algebra over
     the rationals and over the integers.

2.4. Commuting  hyperbolic automorphisms of the torus. Genuine higher rank
     condition.  Examples: Cartan actions, symplectic actions.

2.5. Partially hyperbolic  actions by  toral automorphisms. Examples of
     genuinely partially hyperbolic actions. Dimension restrictions.
     Peculiarity of dynamical behavior.

2.6. Background on nilpotent  groups and nilpotent Lie groups. Compact
     nilmanifolds. Examples of hyperbolic actions by automorphisms of
     nilmanifolds.

2.7. First examples of homogeneous  actions on  factors of simple Lie
     groups. Classical geodesic and horocycle flows on surfaces of constant 
     negative curvature as homogeneous flows on  factors of SL(2,R).

2.8. The central example in the theory of higher rank abelian group
     actions: left  translations  by the diagonals on facts of SL(n,R) 
     for n>2 - the Weyl chamber flow.

2.9. Contrast between dynamical properties  of the geodesic flow (n=2) and
     Weyl chamber flow (n>2).


3. ELEMENTS OF RIGIDITY THEORY (approx. two lectures)

3.1. Structural stability and differentiable rigidity. Proof of structural
     stability of hyperbolic automorphisms of the torus. Infinitely many moduli
     for differentiable conjugacy.

3.2. Invariant measures for hyperbolic automorphisms and for higher-rank
     actions. Furstenberg times 2, times 3 problem.

3.3. Model problem in differentiable rigidity: local rigidity for Cartan
     action (two commuting hyperbolic automorphisms) on three-dimensional
     torus.

3.4. Model problem in measure rigidity: the only positive entropy ergodic
     measure for Cartan action on three-dimensional torus is Lebesgue.

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APPLICATION FORM:

ALGEBRAIC ACTIONS OF HIGHER RANK ABELIAN GROUPS AND
INTRODUCTION TO RIGIDITY
 
Feb/March, 2007

Name:

E-mail:

University:

Status: PhD student/MSc student/Postdoc (please choose)

If you are a PhD/MSc student, please provide a 
recommendation letter from your supervisor

Please sketch your background in the topic of the course 
and/or enclose your scientific CV.

Please fill out the form and send it before the application deadline of 22 January 2007 to the e-mail address.

ALGEBRAIC ACTIONS OF HIGHER RANK ABELIAN GROUPS AND INTRODUCTION TO RIGIDITY