Séance Séminaire

Séminaire Gaston Darboux

Metric spectral theory

Metric functionals, a variant of horofunctions, play the role for metric spaces what linear functionals do for linear spaces. They give a notion of weak topology and weak compactness. I will discuss a "metric spectral principle" I observed in 1999 in its first version. Thanks to progress in the determinations of horofunctions for various metric spaces, a fair amount of applications have emerged such as (extensions of): the Wolff-Denjoy theorem in complex analysis, the Carleman-von Neumann mean ergodic theorem, Thurston's spectral theorem for surface homeomorphisms, classification of isometries of non-locally compact hyperbolic spaces, and non-linear Perron-Frobenius results. Thanks to a more recent collaboration with S. Gouëzel, a significant strengthening of the subadditive ergodic theorem was established and yielded a random version of this spectral principle. This has applications to random walks, multiplicative ergodic theorems, law of large numbers for random variables with exotic moment conditions, and random difference equations,