### Minisemester

Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory

#### June 27 – July 24, 2011, Będlewo

The main aim of the summer school is to provide to participants modern methods of dynamical systems oriented to applications in analysis, geometry and number theory. This area is one of best developing in modern mathematics; one of the leaders of this area, **Elon Lindenstrauss**, was awarded **Fields Medal **during the last ICM (India, Hydebarad). The concluding workshop will sum up the current scientific knowledge of the subject.

First week courses are aimed at the beginning and middle years Ph.D. students who have good knowledge of real analysis and basic topology but may not have sufficient familiarity with basics of dynamics and the theory of Lie groups. The goal is to bring participants at the level necessary to participate in the advanced courses.

In addition to the participants of week 1 advanced courses are aimed at advanced graduate students, your researchers working in dynamical systems and related areas as well as to mathematicians of all ages interested in various aspects of the subjects. There is no requirement to take all advanced courses simultaneously. In fact it is expected that each participant will participate on the average in four courses out of six.

Financial support toward local expenses for a restricted number of participants will be available.

The minisemester is supported by Stefan Banach International Mathematical Center (part of the Institute of Mathematics of the Polish Academy of Sciences), Center for Dynamics and Geometry at Penn State University, National Science Foundation and Nicolaus Copernicus University.

### Organizing Committee

- Mariusz Lemańczyk (Uniwersytet M. Kopernika), Chair, mlem at mat.umk.pl
- Krzysztof Frączek (Uniwersytet M. Kopernika), fraczek at mat.umk.pl
- Andrey Gogolev (University of Texas, Austin), agogolev at math.utexas.edu
- Anatole Katok (Penn State University), katok_a at math.psu.edu

### Scientific Committee

- Anatole Katok (Penn State University), Chair
- Manfred Einsiedler (EHT, Zurich)
- Giovanni Forni (University of Maryland)
- Mariusz Lemańczyk (Uniwersytet M. Kopernika)
- Francois Ledrappier (University of Notre Dame)
- Federico Rodriguez Hertz (IMERL, Montevideo)
- Klaus Schmidt (University of Vienna)
- Bassam Fayad (CNRS, Paris)
- Chengbo Yue (Math. Institute, Chinese Academy of Sciences)

### School courses

**TUTORIALS, Week 1**

** Alexander Gorodnik:**Basics of Lie theory, algebraic and arithmetic groups for dynamicists

**Francois Ledrappier: **

Topics on entropy, **(****lecture notes)**

**Omri Sarig**:

Introduction to ergodic theory

*Each course is expected to run daily with a double lecture: two parts 45-50 minutes with a break.*

*It is planned a seminar run by A. Katok in the evenings during the first week to help tying loose ends.*

**Auxiliary lecture notes**:

M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in dynamics;

A. Katok and V. Climenhaga, Measure theory through dynamical eyes;

A. Katok and V. Climenhaga, Basic material on hyperbolic geometry from the book *Lectures on Surfaces: (almost) everything you wanted to know about them*, published by AMS in 2008;

### ADVANCED COURSES

** Manfred Einsiedler:**Homogeneous dynamics and applications in number theory

Week 2

**Auxiliary lecture notes**:M. Einsiedler and T. Ward, More Dynamics on Quotients of the Hyperbolic Plane, Chapter 11 from the recently published book

*Ergodic Theory with a view towards Number Theory,*Graduate Texts in Mathematics, Vol. 259, Springer-Verlag, London, 2011;

M. Einsiedler, RATNER'S THEOREM ON SL(2,R)-INVARIANT MEASURES;

Dave Witte Morris,

*Ratner’s Theorems*

*on Unipotent Flows;*

**Livio Flaminio**:

Introduction to the theory of unitary group representations and its applications to dynamics **(lecture notes 1)**

Weeks 2-3**Giovanni Forni and Carlos Matheus**:

Introduction to Teichmuller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards

Weeks 2-3**Auxiliary lecture notes**:

C. Matheus, *Lyapunov spectrum of the Kontsevich-Zorich cocycle on the Hodge bundle over square-tiled cyclic covers* **(parts I, II, III and IV), **notes on Matheus’ Weblog

**Michael Hochman**:

Interactions between ergodic theory and fractal geometry

Weeks 2-3**Anatole Katok and Zhenqi Jenny Wang**:

Introduction to KAM (Kolmogorov-Arnold-Moser) method and its applications to rigidity of group actions, **(slides)**

Weeks 2-3**Auxiliary lecture notes**:

A. Katok and V. Nitica, Principal classes of algebraic actions, Chapter 2 from the recently published book *RIGIDITY IN HIGHER RANK ABELIAN GROUP ACTIONS I. Introduction and cocycle problem*, Cambridge University Press 2011;

A. Katok and D. Damjanovic, Local rigidity of partially hyperbolic actions. I. KAM method and **Z**^{k } actions on the Torus, *Annals of Mathematics, ***172** (2010), 1805-1858

**Federico Rodriguez Hertz**:

Measure rigidity for group actions

Week 2**Auxiliary lecture notes**:

B. Kalinin and A. Katok, a part of the article Invariant measures for actions of higher rank abelian groups, *in Smooth Ergodic Theory and its applications, Proc. Symp. Pure Math.*, **69** (2001), 593-637

### Workshop Programme

- Alexander I. Bufetov:
*On the Vershik-Kerov Conjecture Concerning the Shannon-McMillan-Breiman Theorem for the Plancherel Family on Measures on the Space of Young Diagrams* - Van Cyr:
*A number theoretic question arising in the geometry of plane curves and in billiard dynamics* - Vincent Delecroix:
*Diffusion in the periodic wind-tree model* - Alexander Gorodnik:
*Mixing for higher-rank abelian actions on nilmanifolds* - Mike Hochman:
*Slow entropy invariants and differentiable models for infinite-measure preserving $Z^k$ actions* - Huyi Hu:
*Essential coexistence of chaotic and nonchaotic behavior in a volume preserving diffeomorphism* - Shirali Kadyrov:
*Entropy and escape of mass for diagonal actions* - Anatole Katok:
*The new notion of entropy for actions of higher rank abelian groups, and its connections to slow entropy and rigidity* - Svetlana Katok:
*Reduction theory, coding of geodesics, and continued fractions* - Felipe Ramirez:
*Higher cohomology for Anosov actions* - Federico Rodriguez Hertz:
*Arithmeticity for some higher rank abelian actions* - Bryce Weaver:
*Growth Rate of Periodic Orbits for a Class of Non-Uniformly Hyperbolic Flows.*

### Participants

- Przemysław Berk (Nicolaus Copernicus University)
- Dmitry Bolotov (National Academy of Sciences of Ukraine)
- Michael Bromberg (Tel-Aviv University)
- Alexander BUFETOV (Steklov Institute and Rice University)
- Aline Cerqueira (IMPA - Institute for Pure and Applied Mathematics)
- Gonzalo Contreras (CIMA)
- Van Cyr (Northwestern University)
- Vincent Delecroix (Institut de Mathématiques de Luminy)
- Eugeniusz Dymek (Nicolaus Copernicus University)
- Manfred Einsiedler (ETH Zürich)
- Alexander Felshtyn (University of Szczecin)
- Livio Flaminio (Université Lille 1)
- Giovanni Forni (University of Maryland)
- Krzysztof Frączek (Nicolaus Copernicus University)
- Światosław Gal (Uniwersytet Wrocławski and Universität Wien)
- Oleksandr Gomilko (Nicolaus Copernicus University)
- Alexander Gorodnik (University of Bristol)
- Eugene Gutkin (Nicolaus Copernicus University)
- Mike Hochman (Princeton University)
- Huyi Hu (Michigan State University)
- Dawid Huczek (Wrocław University of Technology)
- Evgeny Iashagin (Federal University of Kazan)
- Joanna Jaroszewska (University of Warsaw)
- Ludwik Jaksztas (Warsaw University of Technology)
- Shirali Kadyrov (University of Bristol)
- Charlene Kalle (University of Vienna)
- Piotr Kamieński (Jagiellonian University)
- Olena Karpel (Institute for Low Temperature Physics of NAS)
- Anatole Katok (Penn State University)
- Svetlana Katok (Penn State University)
- Alena Khmelnitckaia (Federal University of Kazan)
- Stefan Kotowski (Polsko Japońska Wyzsza Szkoła Technik Komputerowych)
- Joanna Kułaga (Nicolaus Copernicus University)
- Michal Kupsa (Institute of Information Theory and Automation of the ASCR)
- Viktoriia Lebid (Institute of Mathematics NAS of Ukraine)
- Francois Ledrappier (Universite Paris 6)
- Mariusz Lemańczyk (Nicolaus Copernicus University)
- Pierre Lofredi (Institut de Mathématiques de Luminy)
- Beverly Lytle (ETH Zürich)
- Alex Maier (ETH Zurich)
- Carlos Matheus Silva Santos (CNRS - LAGA - Université Paris 13)
- Mykola Matviichuk (National Taras Shevchenko University of Kyiv)
- Luke Mohr (University of Massachusetts Amherst)
- Anke Pohl (ETH Zurich)
- Raquel Portela (University of the State of Rio de Janeiro)
- Alexander Prikhodko (Lomonosov Moscow State University)
- Felipe Ramirez (University of Bristol)
- Henry Reeve (University of Bristol)
- Damoon Robatian (National Academy of Science of Ukraine)
- Federico Rodriguez Hertz (Penn State University)
- Romulo Rosa (Universidade Federal Fluminense)
- Rene Ruehr (ETH Zurich)
- Omri Sarig (Weizmann Institute of Science)
- Tarek SELLAMI (Institut de mathématiques de Luminy)
- Jacek Serafin (Wrocław University of Technology)
- Enhui Shi (Suzhou University)
- Ronggang Shi (Xiamen University)
- Justyna Signerska (Polish Academy of Sciences)
- Anton Solomko (Institute for Low Temperature Physics of NAS)
- Maxim Solund Kirsebom (University of Bristol)
- Gabriel Strasser (University of Vienna)
- Pablo Suárez Serrato (Universidad Politecnica de Catalunya)
- Rober Szczelina (Jagiellonian University)
- Anna Szymusiak (Jagiellonian University)
- James Tanis (University of Maryland)
- Pavlo Tsytsura (Pennsylvania State University)
- Corinnna Ulcigrai (University of Bristol)
- Edson Vargas (Sao Paulo University)
- Kurt Vinhage (Pennsylvania State University)
- Zhenqi Wang (Yale University)
- Bryce Weaver ( )
- Piotr Woronowicz (Nicolaus Copernicus University)
- Weisheng Wu (Pennsylvania State University)
- Yuqing Zhang (University of Vienna)
- Charalampos Zinoviadis (University of Turku)