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SUMMARY:Summer School in Dynamics (Advanced | (smr 3253)
DTSTART;VALUE=DATE-TIME:20180723T060000Z
DTEND;VALUE=DATE-TIME:20180727T200000Z
DTSTAMP;VALUE=DATE-TIME:20181114T215045Z
UID:indico-event-8467@ictp.it
DESCRIPTION:\nWeek 1 (see smr3226)\n\nA circle of concepts and methods in
dynamics\n\nBasic concepts in dynamics will be introduced\, with many exam
ples\, especially in the setting of circle maps. Topics include rotations
of the circle\, doubling map\, Gauss map and continued fractions and an in
troduction 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.\n\nWeek 2\n\nErgodicity in smooth dynamics
(10h\, Jana Rodriguez-Hertz and Amie Wilkinson) \n\nThe concept of ergodic
ity is a central hypothesis in statistical mechanics\, one whose origins c
an be traced to Boltzmann's study of ideal gases in the 19th century. Loo
sely speaking\, a dynamical system is ergodic if it does not contain any p
roper 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 fir
st stepping stone in a path through the study of statistical properties of
dynamical systems\, a field known as Ergodic Theory.\n\nWe will develop t
he ergodic theory of smooth dynamical systems\, starting with the fundamen
tal\, linear examples of rotations and doubling maps on the circle introdu
ced in Week 1. We will develop some tools necessary to establish ergodici
ty of nonlinear smooth systems\, such as those investigated by Boltzmann a
nd PoincarÃ© in the dawn of the subject of Dynamical Systems. Among these
tools are distortion estimates\, density points\, invariant foliations an
d absolute continuity. Closer to the end of the course\, we will focus o
n the ergodic theory of Anosov diffeomorphisms\, an important family of "t
oy models" of chaotic dynamical systems.\n\n\nRenormalization in entropy z
ero systems (5h\, Corinna Ulcigrai)\n\nRotations of the circle are perhaps
the most basic examples of low complexity (or "entropy zero") dynamical s
ystems. A key idea to study systems with low complexity is renormalization
. The Gauss map and continued fractions can be seen as a tool to renormali
ze rotations\, i.e.study the behaviour of a rotation on finer and finer sc
ales. We will see two more examples of renormalization in action.\n\n The
first is the characterization of Sturmian sequences\, which arise as symb
olic coding of trajectories of rotations (and hint at more recent developm
ents\, 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 so
me ideas of how it can be used (in some simplified settings) to study inva
riant measures and (unique) ergodicity and deviations of ergodic averages
for IETs. \n\n------------------------------------------------------------
--------------------------------\nTutorial and exercise sessions will be h
eld regularly and constitute an essential part of the school. \nTutors: Ol
iver BUTTERLEY (ICTP)\, Irene PASQUINELLI (Durham University\, UK)\, David
e RAVOTTI\, (University of Bristol\, UK)\, Lucia SIMONELLI (ICTP)\, Kadim
WAR (Ruhr-UniversitÃ¤t\, Bochum\, Germany).\n\nWomen in Mathematics: Activ
ities directed to encourage and support women in mathematics\, such as pan
el discussions and small groups mentoring and networking\, will be organiz
ed during the event.\n\n\nhttp://indico.ictp.it/event/8467/
LOCATION:ICTP Budinich Lecture Hall (LB)
URL:http://indico.ictp.it/event/8467/
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