Random dynamics on real and complex projective surfaces

Serge Cantat (1) , Romain Dujardin (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We initiate the study of random iteration of automorphisms of real and complex projective surfaces, or more generally compact Kähler surfaces, focusing on the fundamental problem of classification of stationary measures. We show that, in a number of cases, such stationary measures are invariant, and provide criteria for uniqueness, smoothness and rigidity of invariant probability measures. This involves a variety of tools from complex and algebraic geometry, random products of matrices, non-uniform hyperbolicity, as well as recent results of Brown and Rodriguez Hertz on random iteration of surface diffeomorphisms.

Domaines: Systèmes dynamiques [math.DS]; Géométrie algébrique [math.AG]

Serge Cantat, Romain Dujardin. Random dynamics on real and complex projective surfaces. Journal für die reine und angewandte Mathematik, 2023, ⟨10.1515/crelle-2023-0038⟩. ⟨hal-02780876v3⟩