The goal of this series of lecture is to present the relation between the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.
In the first talk, we will review the dynamics of germs $f:({\mathbb C},0)\to ({\mathbb C},0)$, in...
(Joint work with Xavier Buff) Given a meromophic connection with a pole of degree>1 near a puncture of a Riemann surface, we introduce a sequence of asymptotic values and use it to define an invariant that allows for a complete local classification of those objects, up to local conformal isomorphism. We also provide a geometric description.
The goal of this series of lecture is to present the relation between the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.
In the second lecture, we will explain how, to each germ $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$...
A meromorphic one form with poles on a Riemann surface defines naturally a translation surface of infinite area. In this talk, after seeing briefly how such structures appear naturally when studying usual translation surfaces, we will describe the orbits of the geodesic flow and show how we can use this result to classify the connected components of the corresponding moduli space.
The goal of this series of lecture is to present the relation between the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.
In the third lecture, we will study the geodesic flow of meromorphic affine surfaces modeled on compact...
Through explicit examples introduced by Samuele Mongodi and myself, we will see how the resolution of singularities of vector fields of McQuillan and Panazzolo, and the resolution along separatrices of Lopez-Hernanz, Ribon, Sanz-Sanchez and Vivas, intervene in the study of parabolic manifolds for tangent to the identity germs in dimension 3.
Part of the talk is based on a work in progress...
The goal of this series of lecture is to present the relation between the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.
In the fourth lecture, we will explain how, using the dynamics of the geodesic flow on affine surfaces,...
Translation surfaces are (very) particular type of affine surfaces where transition maps are translations. Though, any affine surface admits a cover which is a translation surface (possibly of infinite type). The goal of my talk is to introduce a metric invariant for singularities of infinite type translation surfaces due to Bowman-Valdez and explain how it is related to the completeness of...
Parabolic implosion is a tool for studying the dynamics of perturbations of a map with a fixed point tangent to the identity, or more generally with at least one eigenvalue which is a root of unity. We will start by surveying classical parabolic implosion in dimension one, and then we will explain an ongoing work on parabolic implosion of germs tangent to the identity in dimension 2.
Joint...
The goal of this series of lecture is to present the relation between the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.
In the last lecture, we will try to formulate open problems concerning the dynamics of germs...
