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SUMMARY:Summer School in Dynamics (Introductory and Advanced) | (smr 3226)
DTSTART;VALUE=DATE-TIME:20180716T060000Z
DTEND;VALUE=DATE-TIME:20180727T200000Z
DTSTAMP;VALUE=DATE-TIME:20180718T104334Z
UID:indico-event-8325@ictp.it
DESCRIPTION:\n\n\nWeek 1A circle of concepts and methods in dynamics.\n \
nBasic concepts in dynamics will be introduced\, with many examples\, espe
cially in the setting of circle maps. Topics include rotations of the circ
le\, 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 a
nd renormalization.\n \n \nWeek 2\n Ergodicity in smooth dynamics (10h\
, Jana Rodriguez-Hertz and Amie Wilkinson) \n \nThe 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. Loose
ly speaking\, a dynamical system is ergodic if it does not contain any pro
per subsystem\, where the notion of "proper" is defined using measures.
A powerful theorem of Birkhoff from the 1930's states that ergodicity is e
quivalent 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 a
verage value over the entire space. The property of ergodicity is the firs
t stepping stone in a path through the study of statistical properties of
dynamical systems\, a field known as Ergodic Theory.\n \nWe will develop
the ergodic theory of smooth dynamical systems\, starting with the fundame
ntal\, linear examples of rotations and doubling maps on the circle introd
uced in Week 1. We will develop some tools necessary to establish ergodi
city of nonlinear smooth systems\, such as those investigated by Boltzmann
and Poincaré in the dawn of the subject of Dynamical Systems. Among th
ese tools are distortion estimates\, density points\, invariant foliations
and absolute continuity. Closer to the end of the course\, we will fo
cus on the ergodic theory of Anosov diffeomorphisms\, an important family
of "toy models" of chaotic dynamical systems.\n \n Renormalization in en
tropy zero systems (5h\, Corinna Ulcigrai)\n \nRotations of the circle ar
e perhaps the most basic examples of low complexity (or "entropy zero") dy
namical systems. A key idea to study systems with low complexity is renorm
alization. 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 actio
n.\n \n The first is the characterization of Sturmian sequences\, which
arise as symbolic coding of trajectories of rotations (and hint at more re
cent developments\, such as the characterization of cutting sequences for
billiards in the regular octagon). The second concerns interval exchange m
aps (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 ergo
dic averages for IETs.\n \n----------------------------------------------
-------------------------------------------\n \nTutorial and exercise ses
sions will be held regularly and constitute an essential part of the schoo
l.\nTutors: Oliver BUTTERLEY (ICTP)\, Irene PASQUINELLI (Durham University
\, UK)\, Davide RAVOTTI\, (University of Bristol\, UK)\, Lucia SIMONELLI (
ICTP)\, Kadim WAR (Ruhr-Universität\, Bochum\, Germany).\n \nWomen in Ma
thematics: Activities directed to encourage and support women in mathemati
cs\, such as panel discussions and small groups mentoring and networking\,
will be organized during the event.\n\nhttp://indico.ictp.it/event/8325/
LOCATION:ICTP Budinich Lecture Hall (LB)
URL:http://indico.ictp.it/event/8325/
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