| Description |
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. |
COURSE MATERIAL
Summer School in Dynamics (Introductory and Advanced) | (smr 3226)
Go to day
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08:30 - 18:30
Location: Leonardo Building - Budinich Lecture Hall -
08:30
REGISTRATION
30' (
Leonardo Building - Main Lobby
)
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09:00
Welcome and Opening
10'
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09:10
Rotations of the circle and renormalization 1
50'
Speaker: Corinna Ulcigrai (University of Bristol, UK) Material: 
Video
notes
- 10:00 Break 20'
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10:20
Piecewise expanding maps of the circle 1
50'
Speaker: Jana Rodriguez-Hertz (Southern University of Science and Technology, Shenzhen, China) Material:
Slides
Video
- 11:10 Break 20' ( )
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11:30
Introduction to ergodic theory 1
50'
Speaker: Oliver Butterley, ICTP Material:
Slides
Video
notes
- 12:20 Lunch break 1h40'
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14:00
Exercise Sessions
1h30'
Material:
notes
- 16:00 Coffee break 30' ( Leonardo Building Terrace )
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16:30
Socializing Activities
1h0' (
Leonardo Building Terrace
)
- 17:30 Get-together Drink 1h0' ( Leonardo Building Terrace )
-
08:30
REGISTRATION
30' (
Leonardo Building - Main Lobby
)
-
08:30 - 18:30
-
-
09:10 - 17:30
Location: Leonardo Building - Budinich Lecture Hall -
09:10
Rotations of the circle and renormalization 2
50'
Speaker: Corinna Ulcigrai (University of Bristol, UK) Material:
Video
notes
- 10:00 Break 20'
-
10:20
Piecewise expanding maps of the circle 2
50'
Speaker: Jana Rodriguez-Hertz (Southern University of Science and Technology, Shenzhen, China) Material:
Slides
Video
- 11:10 Break 20'
-
11:30
Introduction to ergodic theory 2
50'
Speaker: Irene Pasquinelli, Durham Univ. Material:
Video
notes
- 12:20 Lunch break 1h40'
-
14:00
Exercise Sessions
1h30'
Material:
notes
- 16:00 Coffee break 30'
-
16:30
Small groups discussion for women and gender minorities
1h0' (
Leonardo Building Terrace
)
-
09:10
Rotations of the circle and renormalization 2
50'
-
09:10 - 17:30
-
-
09:10 - 18:00
Location: Leonardo Building - Budinich Lecture Hall -
09:10
Introduction to ergodic theory 3
50'
Speaker: Stefano Luzzatto, ICTP Material:
Video
- 10:00 Break 20'
-
10:20
Piecewise expanding maps of the circle 3
50'
Speaker: Jana Rodriguez-Hertz Material:
Slides
Video
- 11:10 Break 20'
-
11:30
Rotations of the circle and renormalization 3
50'
Speaker: Corinna Ulcigrai (University of Bristol, UK) Material:
Video
notes
- 12:20 Lunch break 1h40'
-
14:00
Exercise Sessions
1h30'
Material:
notes
- 16:00 Coffee break 30' ( Leonardo Building Terrace )
-
16:30
ICTP and OWSD Presentation
1h30' (
Leonardo Building - Budinich Lecture Hall
)
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09:10
Introduction to ergodic theory 3
50'
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09:10 - 18:00
-
-
09:10 - 17:30
Location: Leonardo Building - Budinich Lecture Hall -
09:10
Piecewise expanding maps of the circle 4
50'
Speaker: Jana Rodriguez-Hertz Material:
Slides
Video
- 10:00 Break 20'
-
10:20
Rotations of the circle and renormalization 4
50'
Speaker: Corinna Ulcigrai Material:
Video
- 11:10 Break 20'
-
11:30
Introduction to ergodic theory 4
50'
Speaker: Lucia Simonelli, ICTP Material:
Video
- 12:20 Lunch break 1h40'
-
14:00
Exercise Sessions
1h30'
- 16:00 Coffee break 30' ( Leonardo Building Terrace )
-
16:30
Career Panel
1h0'
-
09:10
Piecewise expanding maps of the circle 4
50'
-
09:10 - 17:30
-
-
09:10 - 16:30
Location: Leonardo Building - Budinich Lecture Hall -
09:10
Introduction to ergodic theory 5
50' (
Leonardo Building - Budinich Lecture Hall
)
Speaker: Davide Ravotti, Univ. of Bristol Material:
Video
- 10:00 Break 20'
-
10:20
Rotations of the circle and renormalization 5
50'
Speaker: Corinna Ulcigrai Material:
Video
- 11:10 Break 20'
-
11:30
Piecewise expanding maps of the circle 5
50'
Speaker: Jana Rodriguez-Hertz Material:
Slides
Video
- 12:20 Lunch break 1h40'
-
14:00
Exercise Sessions
1h30' (
Leonardo Building - Budinich Lecture Hall
)
- 16:00 Coffee break 30'
-
09:10
Introduction to ergodic theory 5
50' (
Leonardo Building - Budinich Lecture Hall
)
-
09:10 - 16:30