A Summer School in Dynamical Systems which will be held at ICTP, Trieste (Italy) on 16-27 July 2018.

The school is targeted to Masters and beginning PhD students and aims at introducing a variety of fundamental ideas and techniques in dynamics. The first week will not assume prior knowledge in the field of dynamical systems. Students can also register for the second week only.

Female students are especially encouraged to apply. A limited number of grants are available to support the attendance of selected participants, with priority given to participants from developing countries.

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**Organizers and lectures:**

-Jana RODRIGUEZ-HERTZ (Southern University of Science and Technology of China)

-Corinna ULCIGRAI (University of Bristol, UK)

-Amie WILKINSON (University of Chicago, USA)

Local organizer: Stefano Luzzatto (ICTP)

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**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.

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**Week 2**

__(10h, Jana Rodriguez-Hertz and Amie Wilkinson)__

*Ergodicity in smooth dynamics*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.

*(5h, Corinna Ulcigrai)*

__Renormalization in entropy zero systems__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.

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**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).

__Activities directed to__

**Women in Mathematics:**__encourage and support women in mathematics__, such as panel discussions and small groups mentoring and networking, will be organized during the event.

**Deadlines for applications:**

2 April 2018 (for applicants requesting financial support)

25 June 2018 (for applicants NOT requesting financial support)

Please note the fast-approaching DEADLINE (2 April 2018) for applicants requesting financial support!