Peter Haïssinsky (Université Paul Sabatier, Toulouse)
Some topological characterizations of rational maps and Kleinian groups
Abstract:
This course has the ambition to present methods coming from quasiconformal geometry in metric spaces in order to characterize conformal dynamical systems. We will focus on some specific classes of rational maps and of Kleinian groups (semi-hyperbolic rational maps and convex-cocompact Kleinian groups). These classes can be characterized among conformal dynamical systems by topological properties which will enable us to define classes of topological dynamical systems on the sphere (coarse expanding conformal maps and uniform convergence groups). It turns out that these classes carry some non trivial geometric information enabling to associate a coarse conformal structure invariant by their dynamics . This conformal structure will be derived from hyperbolic geometry in the sense Gromov. We associate to this conformal structure a numerical invariant, the Ahlfors regular conformal dimension, which will enable to characterize genuine conformal dynamical systems.