Scientific Calendar Event

 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.

### Material

Go to day
• Monday, 16 July 2018
• 08:30 - 18:30  Location: Leonardo Building - Budinich Lecture Hall
• 08:30 REGISTRATION 30' ( Leonardo Building - Main Lobby )
• 09:00 Welcome and Opening 10'
• 09:10 Rotations of the circle and renormalization 1 50'  Speaker: Corinna Ulcigrai (University of Bristol, UK) Material:
• 10:00 Break 20'
• 10:20 Piecewise expanding maps of the circle 1 50'  Speaker: Jana Rodriguez-Hertz (Southern University of Science and Technology, Shenzhen, China) Material:
• 11:10 Break 20' ( )
• 11:30 Introduction to ergodic theory 1 50'  Speaker: Oliver Butterley, ICTP Material:
• 12:20 Lunch break 1h40'
• 14:00 Exercise Sessions 1h30'  Material:
• 16:00 Coffee break 30' ( Leonardo Building Terrace )
• 16:30 Socializing Activities 1h0' ( Leonardo Building Terrace )
• 17:30 Get-together Drink 1h0' ( Leonardo Building Terrace )
• Tuesday, 17 July 2018
• 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:
• 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:
• 11:10 Break 20'
• 11:30 Introduction to ergodic theory 2 50'  Speaker: Irene Pasquinelli, Durham Univ. Material:
• 12:20 Lunch break 1h40'
• 14:00 Exercise Sessions 1h30'  Material:
• 16:00 Coffee break 30'
• 16:30 Small groups discussion for women and gender minorities 1h0' ( Leonardo Building Terrace )
• Wednesday, 18 July 2018
• 09:10 - 18:00  Location: Leonardo Building - Budinich Lecture Hall
• 09:10 Introduction to ergodic theory 3 50'  Speaker: Stefano Luzzatto, ICTP Material:
• 10:00 Break 20'
• 10:20 Piecewise expanding maps of the circle 3 50'  Speaker: Jana Rodriguez-Hertz Material:
• 11:10 Break 20'
• 11:30 Rotations of the circle and renormalization 3 50'  Speaker: Corinna Ulcigrai (University of Bristol, UK) Material:
• 12:20 Lunch break 1h40'
• 14:00 Exercise Sessions 1h30'  Material:
• 16:00 Coffee break 30' ( Leonardo Building Terrace )
• 16:30 ICTP and OWSD Presentation 1h30' ( Leonardo Building - Budinich Lecture Hall )
• Thursday, 19 July 2018
• 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:
• 10:00 Break 20'
• 10:20 Rotations of the circle and renormalization 4 50'  Speaker: Corinna Ulcigrai Material:
• 11:10 Break 20'
• 11:30 Introduction to ergodic theory 4 50'  Speaker: Lucia Simonelli, ICTP Material:
• 12:20 Lunch break 1h40'
• 14:00 Exercise Sessions 1h30'
• 16:00 Coffee break 30' ( Leonardo Building Terrace )
• 16:30 Career Panel 1h0'
• Friday, 20 July 2018
• 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:
• 10:00 Break 20'
• 10:20 Rotations of the circle and renormalization 5 50'  Speaker: Corinna Ulcigrai Material:
• 11:10 Break 20'
• 11:30 Piecewise expanding maps of the circle 5 50'  Speaker: Jana Rodriguez-Hertz Material:
• 12:20 Lunch break 1h40'
• 14:00 Exercise Sessions 1h30' ( Leonardo Building - Budinich Lecture Hall )
• 16:00 Coffee break 30'
• Monday, 23 July 2018
• 09:10 - 19:00  Location: Leonardo Building - Budinich Lecture Hall
• 09:10 Ergodicity in smooth dynamics 1 50'  Speaker: Jana Rodriguez-Hertz and Amie Wilkinson
• 10:00 Break 20'
• 10:20 Ergodicity in smooth dynamics 2 50'  Speaker: Jana Rodriguez-Hertz and Amie Wilkinson
• 11:10 Break 20'
• 11:30 Renormalization in entropy zero systems 1 50'  Speaker: Corinna Ulcigrai (University of Bristol, UK)
• 12:20 Lunch break 1h40'
• 14:00 Exercise Sessions 1h30'
• 16:00 Coffee Break 30'
• 16:30 Poster Session 1h0' ( Leonardo Building - Lobby )
• 17:30 Reception 1h30' ( Leonardo Building - Lobby )
• Tuesday, 24 July 2018
• 09:10 - 16:30  Location: Leonardo Building - Budinich Lecture Hall
• 09:10 Renormalization in entropy zero systems 2 50'  Speaker: Corinna Ulcigrai
• 10:00 Break 20'
• 10:20 Ergodicity in smooth dynamics 3 50'  Speaker: Jana Rodriguez-Hertz and Amie Wilkinson
• 11:10 Break 20'
• 11:30 Ergodicity in smooth dynamics 4 50'  Speaker: Jana Rodriguez-Hertz and Amie Wilkinson
• 12:20 Lunch break 1h40'
• 14:00 Exercise Sessions 1h30'
• 16:00 Coffee Break 30'
• Wednesday, 25 July 2018
• 09:10 - 17:30  Location: Leonardo Building - Budinich Lecture Hall
• 09:10 Ergodicity in smooth dynamics 5 50'  Speaker: Jana Rodriguez-Hertz and Amie Wilkinson
• 10:00 Break 20'
• 10:20 Renormalization in entropy zero systems 3 50'  Speaker: Corinna Ulcigrai
• 11:10 Break 20'
• 11:30 Ergodicity in smooth dynamics 6 50'  Speaker: Jana Rodriguez-Hertz and Amie Wilkinson
• 12:20 Lunch break 1h40'
• 14:00 Exercise Sessions 1h30'
• 16:00 Coffee break 30'
• 16:30 Career Panel 1h0'
• Thursday, 26 July 2018
• 09:10 - 20:00  Location: Leonardo Building - Budinich Lecture Hall
• 09:10 Ergodicity in smooth dynamics 7 50'  Speaker: Jana Rodriguez-Hertz and Amie Wilkinson
• 10:00 Break 20'
• 10:20 Ergodicity in smooth dynamics 8 50'  Speaker: Jana Rodriguez-Hertz and Amie Wilkinson
• 11:10 Break 20'
• 11:30 Renormalization in entropy zero systems 4 50'  Speaker: Corinna Ulcigrai
• 12:20 Lunch break 1h40'
• 14:00 Exercise Sessions 1h30'
• 16:00 Coffee Break 30'
• 17:00 Information Fair 1h0' ( Adriatico Guest House - Terrace )
• 18:30 Dinner Reception 1h30' ( Adriatico Guest House - Terrace )
• Friday, 27 July 2018
• 09:10 - 16:30  Location: Leonardo Building - Budinich Lecture Hall
• 09:10 Renormalization in entropy zero systems 5 50'  Speaker: Corinna Ulcigrai
• 10:00 Break 20'
• 10:20 Ergodicity in smooth dynamics 9 50'  Speaker: Jana Rodriguez-Hertz and Amie Wilkinson
• 11:10 Break 20'
• 11:30 Ergodicity in smooth dynamics 10 50'  Speaker: Jana Rodriguez-Hertz and Amie Wilkinson
• 12:20 Lunch break 1h40'
• 14:00 Exercise Sessions 1h30'
• 16:00 Coffee Break 30'