Conférence GAGC 2016 (en 2017)

Mini-cours : Dynamical degrees of birational transformations of surfaces

Non programmé

2h 15m

CIRM, Luminy

Orateur

Jérémy Blanc

Description

The dynamical degree $\lambda (f)$ of a birational transformation $f$ of a surface measures the exponential growth of the formula that define the iterates of $f$. It allows to study the complexity of the dynamic of $f$. For instance, over the field of complex number, the number $\log (\lambda (f))$ gives a upper bound for the topological entropy. The number $\lambda (f)$ is an algebraic integer of a special kind: a Pisot or Salem number. Its value gives informations on the transformation, for instance on the fact that it is conjugate or not to an automorphism of a projective surface. More generally, I will try to explain in this mini-course how a precise information on the dynamical degree allows to understand the geometry of the transformation, for instance its conjugacy class or its centraliser in the group of all birational transformations. References: -J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), no. 6, 1135–1169 -J. Blanc and S. Cantat, Dynamical degrees of birational transformations of projective surfaces. J. Amer. Math. Soc. 29 (2016), no. 2, 415–471.