Conference - Families of algebraic dynamical systems
Rennes, from June 12th to June 16th
Organization board: Serge Cantat, Christophe Dupont
Scientific board: Matthew Baker, Eric Bedford, Serge Cantat, Christophe Dupont, Mattias Jonsson
- Bertrand Deroin (Ecole Normale Supérieure, Paris) Holomorphic families of representations in SL(2,C)
We will survey some aspect of the theory of holomorphic families representations in SL(2,C):
1. Sullivan's stability theory
2. Bifurcation currents
3. Harmonic measures of complex projective structures
- Charles Favre (Ecole Polytechnique, Palaiseau) Degeneration of rational maps of the Riemann sphere
We shall describe how one can control the dynamics of a meromorphic family of rational maps of the Riemann sphere parameterized by the punctured unit disk as one approaches the puncture. Our analysis is based in a crucial way on the interplay between complex and non-archimedean dynamics. We shall also review how this control can be combined with technics from arithmetic geometry to the description of the special curves in the parameter space that contain infinitely many post-critically finite maps.
- Laura de Marco (Northwestern University, Chicago) Rational maps, elliptic curves, and heights
We will study the geometry and arithmetic of families of rational maps and families of elliptic curves. The focus will be on "canonical height functions", introduced by Tate and Neron around 1960 in the setting of abelian varieties and further developed by Call and Silverman (1993) for algebraic dynamical systems. My aim is to present recent results -- both in the setting of elliptic curves and of rational maps -- and to present open questions inspired by the connections between holomorphic dynamics and arithmetic geometry.
- François Berteloot (Toulouse)
- Simon Brandhorst (Hannover): On the dynamical spectrum of projective K3 surfaces.
- Romain Dujardin (Université Paris 6)
- Alexander Gamburd (City University of New-York)
- Thomas Gauthier (Université de Picardie Jules Verne, Amiens): The support of the bifurcation measure has positive volume
- Martin Hils (Paris)
- Sarah Koch (Ann Harbor): Irreducibility of curves in parameter space: cubic polynomials vs. quadratic rational maps.
- Holly Krieger (Cambridge University)
- Juan Rivera-Letelier (Rochester)
- Thomas Scanlon (Berkeley University)
- Tom Tucker (Rochester): Towards a finite index conjecture for iterated Galois groups
- Junyi Xie (Université de Rennes 1): Invariant pencils for polynomial selfmaps of the affine plane
Simon Brandhorst : On the dynamical spectrum of projective K3 surfaces.
The dynamical degree of a surface automorphism is a Salem number, that is, an algebraic integer lambda>1 which is conjugate to 1/\lamda and all whose other conjugates lie on the unit circle. We prove that for each Salem number lambda of degree at most 20, there is a power lambda^n, n in N, which is the dynamical degree of an automorphism of some projective K3 surface.
Thomas Gauthier : The support of the bifurcation measure has positive volume.
The moduli space M_d of degree d>=2 rational maps can naturally be endowed with a measure mu_bif detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure mu_bif has positive Lebesgue measure. To do so, we establish a general criterion for the conjugacy class of a rational map to belong to the support of mu_bif and we exhibit a "large" set of Collet-Eckmann rational maps which satisfy that criterion. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps. This is a joint work with Matthieu Astorg, Nicolae Mihalache and Gabriel Vigny.
Sarah Koch : Irreducibility of curves in parameter space: cubic polynomials vs. quadratic rational maps
Living inside the space of monic centered cubic polynomials, are the curves S_n, which consist of all polynomials f which possess a superattracting cycle of period n. Recently, Arfeux and Kiwi announced a proof that S_n is irreducible for all n>=1. In this talk, we consider the analogous curves which live in the moduli space of quadratic rational maps. It is currently unknown if these curves are irreducible. We discuss some unexpected challenges that arise in the quadratic rational map setting which are absent in the cubic polynomial setting. This talk is based on joint work with E. Hironaka.
Tom Tucker : Towards a finite index conjecture for iterated Galois groups
Let f be a polynomial over a global field. Let G denote the inverse limits of the Galois groups of f^n, where f^n denotes n-th iterate of f. Boston and Jones have suggested that under reasonable hypotheses, one might hope that G has finite index in the full group of automorphisms on an infinite tree corresponding to roots of iterates f^n when f is quadratic. We will show that their conjecture is true over function fields of characteristic 0, and that it would be a consequence of well-known diophantine conjectures over number fields. We will also treat the case of cubic polynomials, where less is known.
Junyi Xie : Invariant pencils for polynomial selfmaps of the affine plane
With Jonsson and Wulcan, we classify polynomial selfmaps f of the affine plane of that preserve an irreducible pencil of curves at infinity. More generally, we study a more general classification problem, where the invariant pencil is replaced by more general numerical data at infinity.