# Conférence - Familles de Systèmes Dynamiques Algébriques

Rennes, du 12 juin au 16 juin

*Comité d'organisation* : Serge Cantat, Christophe Dupont

*Comité scientifique* : Matthew Baker, Eric Bedford, Serge Cantat, Christophe Dupont, Mattias Jonsson

### Mini-courses :

- 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.

### Talks :

- François Berteloot (Toulouse): Bifurcations within holomorphic families of endomorphisms of P^k
- Simon Brandhorst (Hannover): On the dynamical spectrum of projective K3 surfaces
- Romain Dujardin (Université Paris 6): Degenerations of SL(2,C) representations and Lyapunov exponents
- Alexander Gamburd (City University of New-York): Markov triples and strong approximation
- Thomas Gauthier (Université de Picardie Jules Verne, Amiens): The support of the bifurcation measure has positive volume
- Martin Hils (Paris): Model theory of compact complex manifolds with an automorphism
- Sarah Koch (Ann Harbor): Irreducibility of curves in parameter space: cubic polynomials vs. quadratic rational maps
- Holly Krieger (Cambridge University): Reduction of dynatomic curves
- Juan Rivera-Letelier (Rochester): Hecke and Linnik
- Thomas Scanlon (Berkeley University): Applications of characterizations of skew-invariant varieties
- 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

### Abstracts :

François Berteloot : Bifurcations within holomorphic families of endomorphisms of P^k.

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.

Romain Dujardin : Degenerations of SL(2,C) representations and Lyapunov exponents

The talk is a report of work in progress with Bertrand Deroin and Charles Favre. Let G be a finitely generated group endowed with some probability measure mu and (rho_lambda) be an algebraic family of representations of G into SL(2,C), diverging in the representation space as lambda converges to infinity. Using non-Archimedean techniques, we study the asymptotics of the random product of matrices induced by rho_lambda(G,mu) as lambda converges to infinity. In particular we can describe the growth rate of the Lyapunov exponent in terms of non-Archimedean data.

Alexander Gamburd : Markov triples and strong approximation.

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.

Martin Hils : Model theory of compact complex manifolds with an automorphism

One may develop the model theory of compact complex manifolds (CCM) with a generic automorphism in rather close analogy to what has been done for existentially closed difference fields, in important work by Chatzidakis and Hrushovski, among others. The corresponding first order theory CCMA provides a model-theoretic framework for the study of meromorphic dynamical systems. In the talk, I will present some results from 'geometric model theory' which hold in CCMA (e.g. the Zilber trichotomy for 'finite-dimensional' types). This is joint work with Martin Bays and Rahim Moosa.

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.

Holly Krieger : Reduction of dynatomic curves

Dynatomic curves parametrize n-periodic orbits of a one-parameter family of polynomial dynamical systems. These curves lack the structure of their arithmetic-geometric analogues (modular curves of level n) but can be studied dynamically. Morton and Silverman conjectured a dynamical analogue of the uniform boundedness conjecture (theorems of Mazur, Merel), asserting uniform bounds for the number of rational periodic points for such a family. I will discuss recent work towards the function field version of their conjecture, including results on the reduction mod p of dynatomic curves for the quadratic polynomial family z^2+c.

Juan Rivera-Letelier (Rochester): Hecke and Linnik

I will discuss the equidistribution of Hecke operators of p-adic elliptic curves. The most difficult case, of supersingular elliptic curves, is analyzed using Lubin-Katz theory of the canonical subgroups, and the period map of Serre-Tate's theory on the deformation of fomal groups. The key ingredient is a version of Linnik's equidistribution theorem for a certain p-adic quaternion algebra. This is a joint work with Sebastian Herrero and Ricardo Menares.

Thomas Scanlon : Applications of characterizations of skew-invariant varieties

In work with Medvedev, I classified the skew-invariant subvarieties of so-called split polynomial dynamical systems. Here, a split polynomial dynamical system is one of the form F: A^n to A^n given in coordinates as (x_1,...,x_n) -> (f_1(x_1), f_2(x_2),..., f_n(x_n)) where each f_i is a polynomial in one variable. The "skew" in "skew-invariant" means that we work over a field K equipped with an endomorphism sigma : K -> K. A subvariety V of A^n is skew-invariant if F maps V to V^sigma, the transform of V under sigma. In most applications of our theorem to date, only the case that sigma is the identity is used and the resulting classification of the invariant varieties may be obtained from methods of complex dynamics, as shown by Pakovich. In this lecture, I will speak about two applications which make essential use of the generalization to skew-invariance: a theorem proven jointly with Medvedev and Nguyen that Mahler functions of polynomial type with respect to multiplicatively independent exponents are algebraically independent and a project with Medvedev to extend our classification of (skew-)invariant varieties to what we call triangular dynamical systems (though what have been called skew-products in the literature): algebraic dynamical systems of the form F : A^n to A^n given in coordinates as (x_1,...,x_n) -> (f_1(x_1),f_2(x_1,x_2),...,f_n(x_1,...,x_n)) where f_i is a polynomial in the variables x_1,..., x_i.

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.