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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:20180522T193155Z
UID:indico-event-8467@ictp.it
DESCRIPTION:PARTICIPANTS WISHING TO ATTEND BOTH WEEKS OF THE SCHOOL SHOULD
NOT APPLY FROM THIS PAGE BUT SHOULD APPLY HERE: http://indico.ictp.it/eve
nt/8325/\n Participants wishing to attend only the second week can apply t
hrough the link available in the sidebar on this page.\n \n\n ***\nDEADL
INE for applicants requesting Financial Support: 2 APRIL 2018\nDEADLINE
for applicants NOT requesting Financial Support: 25 JUNE 2018Organizers an
d Lecturers:J. Rodriguez-Hertz\, SUS Tech.\, Shenzan\, ChinaC. Ulcigrai\,
Univ. Bristol\, U.K.A. Wilkinson\, Univ. Chicago\, U.S.A.Local Organizer:
S. Luzzatto\, ICTP\, Trieste\, Italy\n\nWeek 1\n\nA circle of concepts and
methods in dynamics\n\nBasic concepts in dynamics will be introduced\, wi
th many examples\, especially in the setting of circle maps. Topics includ
e rotations of the circle\, doubling map\, Gauss map and continued fractio
ns and an introduction to the basic ideas of symbolic codings and invarian
t measures. At the end of the week we will discuss some simple examples of
structural stability and renormalization.\n\nWeek 2\n\nErgodicity in smoo
th dynamics (10h\, Jana Rodriguez-Hertz and Amie Wilkinson) \n\nThe concep
t of ergodicity is a central hypothesis in statistical mechanics\, one who
se origins can be traced to Boltzmann's study of ideal gases in the 19th c
entury. Loosely speaking\, a dynamical system is ergodic if it does not c
ontain any proper subsystem\, where the notion of "proper" is defined usin
g measures. A powerful theorem of Birkhoff from the 1930's states that er
godicity is equivalent to the property that "time averages = space average
s:" 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 ergodicit
y is the first stepping stone in a path through the study of statistical p
roperties of dynamical systems\, a field known as Ergodic Theory.\n\nWe wi
ll develop the ergodic theory of smooth dynamical systems\, starting with
the fundamental\, linear examples of rotations and doubling maps on the ci
rcle introduced in Week 1. We will develop some tools necessary to establ
ish 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 f
oliations 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.\n\n\nRenormalization
in entropy zero systems (5h\, Corinna Ulcigrai)\n\nRotations 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 ren
ormalization. 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 act
ion.\n\n The first is the characterization of Sturmian sequences\, which a
rise as symbolic coding of trajectories of rotations (and hint at more rec
ent developments\, such as the characterization of cutting sequences for b
illiards in the regular octagon). The second concerns interval exchange ma
ps (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) t
o study invariant measures and (unique) ergodicity and deviations of ergod
ic averages for IETs. \n\n------------------------------------------------
--------------------------------------------\nTutorial and exercise sessio
ns will be held regularly and constitute an essential part of the school.
\nTutors: Oliver BUTTERLEY (ICTP)\, Irene PASQUINELLI (Durham University\,
UK)\, Davide RAVOTTI\, (University of Bristol\, UK)\, Lucia SIMONELLI (IC
TP)\, Kadim WAR (Ruhr-Universität\, Bochum\, Germany).\n\nWomen in Mathem
atics: Activities directed to encourage and support women in mathematics\,
such as panel discussions and small groups mentoring and networking\, wil
l be organized 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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