Surfaces in Luminy
October 3 – 7, 2016
The study of dynamical systems on surfaces goes back to Poincaré and Birkhoff, with motivations from celestial mechanics. Here are three examples of recent lines of research:
(1) The rotation set was introduced in the 80’s; numerous results were obtained recently that reveal the links between the rotation set and the dynamical structure;
(2) The realisation of the Zimmer conjecture on surfaces, by Polterovich and Franks-Handel, has drawn attention of researchers on group actions on surfaces;
(3) The Arnol’d conjecture, which generalizes the famous Poincaré-Birkhoff theorem, has motivated the development of Floer homology in symplectic geometry; on surfaces, this leads to a fruitful dialogue between the symplectic and topological concepts.

The developement of topological tools also allows to describe the dynamics from a global viewpoint, and are also fruitful for studying differentiable systems
Dynamics on surfaces are represented in South America, the United States, Japan, France, Germany, … This conference will give the opportunity to share the latest advances and the new methods of the subject. It is a natural sequel to Surfaces in Montevideo (2012) and Surfaces in Sao Paulo (2014).
Scientific Committee

John Franks (Northwestern University)
Nancy Guelman (University of the Republic of Uruguay)
Andres Koropecki (Federal Fluminense University)
Patrice Le Calvez (Institut de Mathématiques de Jussieu – Paris Rive-Gauche)
Tobias Oertel-Jäger (Friedrich Schiller University Jena)

Organizing Committee

François Béguin (Institut Galilée, Paris 13) 
Sylvain Crovisier (Université Paris-Sud)
Andres Koropecki (Federal Fluminense University)
Frédéric Leroux (Université Pierre-et-Marie-Curie)
Isabelle Liousse (Université Lille I)
Emmanuel Militon (Université Nice Sophia Antipolis)


The ergodic theory of groups of diffeomorphisms of the circle

Abstract: We will review some recent advances and open problems, on the dynamical theory of groups of diffeomorphisms of the circle.  We will more precisely study the relation between the ergodic and topological properties of the action.

Forcing theory for transverse trajectories of surface homeomorphisms

Abstract: Several recent papers on surface dynamics have used transverse foliations and maximal isotopies for homeomorphisms isotopic to the identity as a main tool in their work. In this mini-course we will introduce the basic concepts behind this tool and show a new way o deriving useful dynamical information by means of a forcing procedure. The applications involve ways of obtaining existence of non-contractible periodic points with consequences for rotation sets of toral homeomorphisms, exponential growth of periodic orbits and estimates on topological entropy of maps.


Global dynamics for symmetric planar maps

The almost-Borel structure of surface diffeomorphisms

Super generalized pseudo-Anosov maps

Shub conjecture for smooth self-maps of the sphere

Generic homeomorphisms and rotation sets

On the dynamics of minimal homeomorphisms of T2

A Poincaré Bendixson theorem for translation lines and applications to invariant continua

Orderability and groups of homeomorphisms of the circle

Nontrivial attractor-repellor maps of S2 and rotation numbers

Franks-Misiurewicz conjecture for extensions of irrational rotations

Some dynamical applications of Carathéodory’s prime ends theory

C0 Hamiltonian dynamics and the Arnold conjecture

The Ogasa invariant for homology spheres in dimension 3