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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:20180520T133107Z
UID:indico-event-8325@ictp.it
DESCRIPTION:PARTICIPANTS WISHING TO TAKE PART ONLY IN THE ADVANCED SCHOOL
(SECOND WEEK) SHOULD NOT APPLY FROM THIS PAGE BUT SHOULD APPLY HERE: htt
p://indico.ictp.it/event/8467/\n Participants wishing to participate for b
oth weeks can apply through the link available on the sidebar of this page
.\n \n\n ***\nDEADLINE for applicants requesting Financial Support: 2
APRIL 2018\nDEADLINE for applicants NOT requesting Financial Support: 25 J
UNE 2018Organizers and 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 c
ircle of concepts and methods in dynamics.\n\nBasic concepts in dynamics w
ill be introduced\, with many examples\, especially in the setting of circ
le maps. Topics include rotations of the circle\, doubling map\, Gauss map
and continued fractions and an introduction to the basic ideas of symboli
c codings and invariant measures. At the end of the week we will discuss s
ome simple examples of structural stability and renormalization.\n\n\nWeek
2\n\nErgodicity in smooth dynamics (10h\, Jana Rodriguez-Hertz and Amie W
ilkinson) \n\nThe concept of ergodicity is a central hypothesis in statist
ical mechanics\, one whose origins can be traced to Boltzmann's study of i
deal 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 t
he 1930's states that ergodicity is equivalent to the property that "time
averages = space averages:" that is\, the average value of a function take
n along an orbit is the same as the average value over the entire space. T
he property of ergodicity is the first stepping stone in a path through th
e 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 fundamental\, linear examples of rotations and
doubling maps on the circle introduced in Week 1. We will develop some t
ools necessary to establish ergodicity of nonlinear smooth systems\, such
as those investigated by Boltzmann and Poincaré in the dawn of the subjec
t of Dynamical Systems. Among these tools are distortion estimates\, dens
ity points\, invariant foliations and absolute continuity. Closer to the
end of the course\, we will focus on the ergodic theory of Anosov diffeom
orphisms\, an important family of "toy models" of chaotic dynamical system
s.\n\n\nRenormalization in entropy zero systems (5h\, Corinna Ulcigrai)\n\
nRotations of the circle are perhaps the most basic examples of low comple
xity (or "entropy zero") dynamical systems. A key idea to study systems wi
th low complexity is renormalization. The Gauss map and continued fraction
s 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 o
f renormalization in action.\n\n The first is the characterization of Stur
mian sequences\, which arise as symbolic coding of trajectories of rotatio
ns (and hint at more recent developments\, such as the characterization of
cutting sequences for billiards in the regular octagon). The second conce
rns interval exchange maps (IETs)\, which are generalizations of rotations
. We will introduce the Rauzy-Veech algorithm as a tool to renormalize IET
s. As applications\, we will give some ideas of how it can be used (in som
e simplified settings) to study invariant measures and (unique) ergodicity
and deviations of ergodic averages for IETs. \n\n------------------------
-----------------------------------------------------------------\n\nTutor
ial and exercise sessions will be held regularly and constitute an essenti
al part of the school. \nTutors: Oliver BUTTERLEY (ICTP)\, Irene PASQUINEL
LI (Durham University\, UK)\, Davide RAVOTTI\, (University of Bristol\, UK
)\, Lucia SIMONELLI (ICTP)\, Kadim WAR (Ruhr-Universität\, Bochum\, Germa
ny).\n\nWomen in Mathematics: Activities directed to encourage and support
women in mathematics\, such as panel discussions and small groups mentori
ng and networking\, will be organized during the event.\n\n\nhttp://indico
.ictp.it/event/8325/
LOCATION:ICTP Budinich Lecture Hall (LB)
URL:http://indico.ictp.it/event/8325/
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