[Todos CMAT] Fwd: Summer School in Dynamics

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Vie Mar 23 15:42:11 -03 2018


----- Mensaje reenviado de Stefano Luzzatto <luzzatto en ictp.it> -----
  Fecha: Fri, 23 Mar 2018 13:39:11 +0100
     De: Stefano Luzzatto <luzzatto en ictp.it>
Asunto: Summer School in Dynamics
   Para: dynamics en lists.ictp.it


We would like to bring to your attention the Summer School in  
Dynamical Systems which will be held at ICTP, Trieste (Italy) this  
summer, 16-27 July 2018.
http://indico.ictp.it/event/8325/ <http://indico.ictp.it/event/8325/>

Please note the fast-approaching DEADLINE (2 April 2018)  for  
applicants requesting financial support!

The school is targeted to Masters and beginning PhD students and aims  
at introducing a variety of fundamental ideas and techniques in  
dynamics. The first week will not assume prior knowledge in the field  
of dynamical systems. Students can also register for the second week  

Female students are especially encouraged to apply. A limited number  
of grants are available to support the attendance of selected  
participants, with priority given to participants from developing  

Organizers and lectures:

-Jana RODRIGUEZ-HERTZ   (Southern University of Science and Technology  
of China)
-Corinna ULCIGRAI (University of Bristol, UK)
-Amie WILKINSON (University of Chicago, USA)

Local organizer: Stefano Luzzatto (ICTP)

Week 1

A circle of concepts and methods in dynamics.

Basic concepts in dynamics will be introduced, with many examples,  
especially in the setting of circle maps. Topics include rotations of  
the circle, doubling map, Gauss map and continued fractions and an  
introduction to the basic ideas of symbolic codings and invariant  
measures. At the end of the week we will discuss some simple examples  
of structural stability and renormalization.

Week 2

Ergodicity in smooth dynamics (10h, Jana Rodriguez-Hertz and Amie Wilkinson)

The concept of ergodicity is a central hypothesis in statistical  
mechanics, one whose origins can be traced to Boltzmann's study of  
ideal gases in the 19th century.  Loosely speaking, a dynamical system  
is ergodic if it does not contain any proper subsystem, where the  
notion of "proper" is defined using measures.  A powerful theorem of  
Birkhoff from the 1930's states that ergodicity is equivalent to the  
property that "time averages = space averages:" that is, the average  
value of a function taken along an orbit is the same as the average  
value over the entire space. The property of ergodicity is the first  
stepping stone in a path through the study of statistical properties  
of dynamical systems, a field known as Ergodic Theory.

We will develop the ergodic theory of smooth dynamical systems,  
starting with the fundamental, linear examples of rotations and  
doubling maps on the circle introduced in Week 1.  We will develop  
some tools necessary to establish ergodicity of nonlinear smooth  
systems, such as those investigated by Boltzmann and Poincaré in the  
dawn of the subject of Dynamical Systems.  Among these tools are  
distortion estimates, density points, invariant foliations and  
absolute continuity.   Closer to the end of the course, we will focus  
on the ergodic theory of Anosov diffeomorphisms, an important family  
of "toy models" of chaotic dynamical systems.

Renormalization in entropy zero systems (5h, Corinna Ulcigrai)

Rotations of the circle are perhaps the most basic examples of low  
complexity (or "entropy zero") dynamical systems. A key idea to study  
systems with low complexity is renormalization. The Gauss map and  
continued fractions can be seen as a tool to renormalize rotations,  
i.e.study the behaviour of a rotation on finer and finer scales.  We  
will see two more examples of renormalization in action.

  The first is the characterization of Sturmian sequences, which arise  
as symbolic coding of trajectories of rotations (and hint at more  
recent developments, such as the characterization of cutting sequences  
for billiards in the regular octagon). The second concerns interval  
exchange maps (IETs), which are generalizations of rotations. We will  
introduce the Rauzy-Veech algorithm as a tool to renormalize IETs. As  
applications, we will give some ideas of how it can be used (in some  
simplified settings) to study invariant measures and (unique)  
ergodicity and deviations of ergodic averages for IETs.


Tutorial and exercise sessions will be held regularly and constitute  
an essential part of the school.
Tutors: Oliver BUTTERLEY (ICTP), Irene PASQUINELLI (Durham University,  
UK), Davide RAVOTTI, (University of Bristol, UK), Lucia SIMONELLI  
(ICTP), Kadim WAR (Ruhr-Universität, Bochum, Germany).

Women in Mathematics: Activities directed to encourage and support  
women in mathematics, such as panel discussions and small groups  
mentoring and networking, will be organized during the event.

Deadlines for applications:
2 April 2018 (for applicants requesting financial support)
25 June 2018 (for applicants NOT requesting financial support)
Stefano Luzzatto
Mathematics Section
Abdus Salam International Centre for Theoretical Physics
Strada Costiera 11, 34151 Trieste, Italy
Tel +39 040 2240253
luzzatto en ictp.it

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