# SMS 2017 - Application

## School on contemporary dynamical systems

July 10-21, 2017

**Organizers:** Sylvain Crovisier (Université de Paris-Sud, France), Konstantin Khanin (University of Toronto, Canada), Andrés Navas (University of Santiago Chile), Christiane Rousseau (University of Montreal. Canada), Marcelo Viana (IMPA, Brazil), Amie Wilkinson (University of Chicago, USA)

**Description**

This summer school school will focus on Contemporary Dynamical Systems.

The theory of dynamical systems has witnessed very significant developments in the last decades. This progress was acknowledged by the fact that the field was granted 9 lectures at ICM2014. It is also worth mentioning that most of the work of two 2014 Fields medalists, Artur Avila and Maryam Mirzakhani fits into the field of dynamical systems. The theory of dynamical systems is a very broad area, which has both pure and applied mathematics aspects. In this school, we emphasize two main of recent developments, namely those of partial hyperbolicity on one side, and rigidity, group actions and renormalization on the other side, as well as four other related areas.

**Conférenciers:**

**Sylvain Crovisier**, Université de Paris-Sud (France)

Partial hyperbolicity conservative and dissipative

This course will introduce the partially hyperbolic systems, a class of dynamics that generalizes the uniformly hyperbolic systems and has been intensively studied these last years. One will focus on the construction of examples and on the fundamental tools developed in this subject: invariant manifolds, coherence, accessibility,... These technics will appear in A. Wilkinson's course, which is the continuation of this one. No background on hyperbolic dynamics is required and the course may be completed by an exercise session.

**Konstantin Khanin**, University of Toronto (Canada)

Renormalization

In the last 25 years renormalization became one of the main tools in the theory of dynamical systems. In this mini-course we shall discuss renormalization theory in the simplest setting of circle dynamics, and present results in the cases of diffeomorphisms, critical circle maps, and maps with breaks. The last case corresponds to a particular type of nonlinear interval exchange transformations. We shall also discuss the relation between hyperbolicity of renormalizations and rigidity theory. We are not assuming previous knowledge of the subject. All basic concepts and constructions will be introduced in the course, together with examples illustrating these concepts.

**Patrice Le Calvez**, Université Pierre et Marie Curie, Paris (France)

Maximal isotopies, transverse foliations and orbit forcing theory for surface homeomorphisms

The course deals about the dynamical study of surface homeomorphisms, more precisely those that are isotopic to the identity (for example defined by time dependent vector fields). We will begin by exposing Brouwer's theory about plane homeomorphisms and its equivariant foliated version. Some applications will be discussed. Then we will explain how to construct a forcing orbit theory for such homeomorphisms in analogy with the classical Sharkovski's theorem for maps defined on an interval. Then we will give further very recent applications. No background is needed for this course but some basic notions of dynamical systems and topology of surfaces.

**Kathryn Mann**, University of California, Berkeley (USA)

Rigidity and flexibility for groups acting on the circle

An action of a group G on the circle is rigid if every deformation of the action is as trivial as possible. For actions on S^1, the right notion is “semi-conjugate”. In this mini-course, I’ll introduce and contrast techniques available to study rigidity of group actions on the circle in various settings (C^0, C^1, smooth...), building towards the introduction of new perspectives and techniques to study the space of C^0 actions, Hom(G, Homeo(S^1)).

**Andrés Navas**, University of Santiago (Chile)

Groups and dynamics

In this series of lectures we will see how methods from dynamics can reveal algebraic information on groups provided these groups act nicely on nice spaces. Several particular properties (e.g. amenability, property (T)) will be introduced and discussed, and recent relevant theorems will be sketched (and proved, if possible).

**Enrique Pujals**, IMPA (Brazil)

Dynamics of smooth volume-contracting surfaces diffeomorphisms

We will discuss a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics. For that type of systems we will study the structure of the attractors, the problem of density of periodic points and the dynamical scenarios for systems in the boundary of chaos.

**Jean-François Quint**, Université de Bordeaux (France)

Random walks on groups: limit theorems

In this course, I will present limit theorems for random walks on some non-abelian groups, which are analogues of the classical results for random walks on R. An essential tool is the use of spectral results for operators, which are analogues of objects appearing in the theory of hyperbolic dynamical systems.

**Juan Rivera-Letelier**, University of Rochester (USA)

Thermodynamic formalism of one-dimensional maps

In the last few decades there has been great collective effort to extend the thermodynamic formalism beyond the classical uniformly hyperbolic setting, which was developed by Sinai, Ruelle, and Bowen. The focus of this mini-course will be on the recent progress in the one- dimensional setting, where a complete picture is emerging. After a review of the ergodic theory of smooth one-dimensional maps, we will concentrate on the (non-)existence and uniqueness of equilibrium states, and the recent classification of phase transitions for geometric potentials. If time permits we will describe the various surprising phenomena that occur at criticality. We will emphasize the analogy with statistical mechanics whenever possible.

**Federico Rodrigues-Hertz**, PennState University (USA)

Group actions, rigidity and beyond

In this series of lectures we shall introduce the main basic tools to handle smooth group actions displaying some hyperbolicity. We shall show how to use these tools to prove some rigidity results and discuss further lines of research. The class of group actions we shall discuss comes mostly in 2 flavors, actions of lattices in higher rank groups, and actions on low dimensional manifolds.

**Christiane Rousseau**, Université de Montréal (Canada)

Singularities of analytic dynamical systems depending on parameters

One basic lesson in bifurcation theory is that the bifurcations of highest codimension in families of dynamical systems depending on parameters organize the bifurcation diagram. Among these, are the bifurcations of equilibrium points, which are studied through normal forms. In analytic families of dynamical systems, the changes of coordinates to normal form generically diverge. The first two lectures, elementary, will illustrate through examples the geometric obstructions to the convergence to normal form. The last lectures will introduce to the geometric methods allowing proving theorems of analytic classification of unfoldings of singularities.

**Ferrán Valdez**, Universidad Nacional Autónoma (Mexico)

Geometry and dynamics on infinite type flat surfaces

Infinite type flat surfaces (i.e. with not finitely generated fundamental group) arise while studying classical dynamical systems such as (irrational) polygonal billiards, baker's map or wind-tree models. They also happen naturally when studying fibered hyperbolic three manifolds or homogeneous holomorphic foliations. In this minicourse we will focus on the main geometrical invariants (Veech groups, types of singularities,...) of these objects, explore the dynamics of the geodesic flow (ergodicity, recurrence, invariant measures...) and the echo that finite type translation surfaces have in the infinite type realm.

**Marcelo Viana**, IMPA (Brazil)

Lyapunov exponents

Lyapunov exponents of linear cocycles and smooth dynamical systems describe the system’s behavior at the exponential level. The theory has its roots in Lyapunov’s pioneer work on stability of differential equations and, over the last half century, or so, has grown into a very active research field whose applications permeate extend to such fields as spectral theory and number theory. Some of the outstanding issues are: existence on non- vanishing exponents, simplicity of the Lyapunov spectrum, dependence of the exponents on the underlying system.

We will discuss these issues in the elementary setting of products of 2-by-2 matrices, either random or driven by a deterministic system such as a circle rotation. Whenever possible, our emphasis will be on explicit examples. The student is expected to have some basic knowledge of measure theory, but prior contact with dynamics or ergodic theory is not a requisite.

**Amie Wilkinson**, University of Chicago (USA)

The ergodic hypothesis and its sequelae

The celebrated Ergodic Theorems of George Birkhoff and von Neumann in the 1930's gave rise to a mathematical formulation of Boltzmann's Ergodic Hypothesis in thermodynamics. This reformulated hypothesis has been described by a variety of authors as the conjecture that ergodicity -- a form of randomness of orbit distributions -- should be ``the general case" in conservative dynamics. I will discuss remarkable discoveries in the intervening century that show why such a hypothesis must be false in its most restrictive formulation but still survives in some contexts. In particular we will study the case of conservative partially hyperbolic systems introduced in S. Crovisier's course. In the end, I will begin to tackle the question, "When is ergodicity and other chaotic behavior the general case?"

**Financement:**