02/08: Serge Cantat

Stationary measures on real projective surface

Abstract: Consider a real projective surface $X(\mathbb R)$ , and a group $\Gamma$ acting by algebraic diffeomorphisms on $X(\mathbb R)$ . If $\nu$ is a probability measure on $\Gamma$ , one can randomly and independently choose elements $f_j$ in $\Gamma$ and look at the random orbits $x$ , $f_1(x)$ , $f_2(f_1(x))$ , … How do these orbits distribute on the surface? This is directly related to the classification of stationary measures on $X(\mathbb R)$ . I will describe recent results on this problem, all obtained in collaboration with Romain Dujardin. The main ingredients will be ergodic theory, notably the work of Brown and Rodriguez-Hertz, algebraic geometry, and complex analysis. Concrete geometric examples will be given.