Studying the (closure of the) (semi-)conjugacy class of a given group action on a 1-manifold is interesting from many points of view. Depending on the manifold and/or the differentiability involved, one is faced with problems concerning small denominators, growth of groups / orbits, distortion elements, bounded cohomology, group orderability, etc. In this minicourse we will explore several general results on this topic such as the $C^1$ smoothing via (semi-)conjugacies of small group actions and obstructions in class $C^2$ and higher. We will also explore some of the ideas involved in the proof of the connectedness of the space of $\mathbb{Z}^d$ actions by diffeomorphisms of $C^{1+ac}$ regularity (obtained in collaboration with H. Eynard-Bontemps).[-]

Artur Avila est un mathématicien franco-brésilien, il travaille principalement dans les domaines des systèmes dynamiques et de la théorie spectrale. Il a obtenu la Médaille Fields en 2014.

Fiche Wikipédia : https://fr.wikipedia.org/wiki/Artur_Á...

Interview/réalisation/post-production : Stéphanie Vareilles
Cadrage/son/technique : Guillaume Hennenfent - Le Chromophore

Artur Avila is a Franco-Brazilian mathematician, working mainly in the fields of dynamical systems and spectral theory. He was awarded the Fields Medal in 2014.

Wikipedia entry: https://fr.wikipedia.org/wiki/Artur_Á...

Interview/realization/post-production: Stéphanie Vareilles
Camera: Guillaume Hennenfent - Le Chromophore[-]

The preceding Etats de la recherche on this topic happened in Rennes, 2006, and the organizers of the present edition asked me to make the bridge between these two sessions. My 2006 talks were devoted to the theory of heights and equidistribution theorems for algebraic dynamical systems. I will start from there by presenting the framework allowed by Arakelov geometry, and explaining the recent manuscript of X. Yuan and S.-W. Zhang who provide a birational perspective to these concepts. The theory is a bit complex and technical but I will try to emphasize the parallel between those ideas and the ones that lie at the ground of pluripotential theory in complex analysis, or in the theory of b-divisors in algebraic geometry.[-]

In these lectures, we will examine a series of conjectures about the geometry of preperiodic points for endomorphisms of $ \mathbb{P}^{N}$. Lecture 1 will focus on the Dynamical ManinMumford Conjecture (DMM), formulated by Shouwu Zhang in the 1990s as an extension of the well-known Manin-Mumford Conjecture (which investigated the geometry of torsion points in abelian varieties and was proved in the early 1980s by Raynaud). The DMM aims to classify the subvarieties of $\mathbb{P}^{N}$ containing a Zariski-dense set of preperiodic points. Lectures 2 and 3 will be devoted to conjectures that treat families of maps on $\mathbb{P}^{N}$. One conjecture in particular was inspired by the recently-proved ”Relative Manin-Mumford” theorem of Gao-Habegger for abelian varieties, but the dynamical version turns out to be closely related to the study of dynamical stability and to contain many previously-existing questions/conjectures/results about moduli spaces of maps on $\mathbb{P}^{N}$. These lectures are based on joint work with Myrto Mavraki.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]