Surfaces affines et germes tangents à l'identité

Real-time dynamics of vector fields and affine structures on Riemann surfaces

10 juil. 2023, 09:00

1h

Salle K. Johnson (1R3-1er étage) (Institut de Mathématiques)

Orateur

Julio Rebelo

Description

In this talk, we will typically consider a homogeneous polynomial vector field $X$ of degree $\ge 2$ on ${\mathbb{C}}^2$. In particular, the time-one map induced by $X$ defines a germ of parabolic diffeomorphism $h$ of $(C^2,0)$. The vector field $X$ also induces a singular affine structure on the Riemann sphere which, in turn, leads to a "geodesic flow", or "billiard dynamics", encoding much of the dynamics of $h$. We will try to make these connections accurate, in particular explaining how the monodromy of the mentioned affine structure can be read off the projective holonomy of the foliation associated with $X$. If time permits, we might say a word about higher dimensional versions of this construction.