Metric functional analysis and its applications

Abstract: Metric functionals (a variant of Busemann functions) are to metric spaces what linear functionals are to linear spaces. In particular they define a weak topology with compactness properties. Invariant metrics appear in many places of mathematics, and it has turned out that often it is possible to have a description of metric functionals (compare with the determination of dual spaces of Banach spaces). There is one general spectral statement that can be proven. This, when applied to various metric spaces, has many new and old applications, in particular as very specific consequences: the Wolff-Denjoy theorem in complex analysis, the Carleman-von Neumann mean ergodic theorem and Thurston’s spectral theorem for surface homeomorphisms.  It is of interest even in the Banach space setting. Thanks to progress in subadditive ergodic theory, a corresponding ergodic theorem can also be established. The first version appeared in a joint work with Ledrappier, the more general one in joint work with Gouëzel. Applicable to random walks on basically any group it addresses a vaguely formulated question of Furstenberg from 1963. In the particular setting of invertible matrices and symmetric spaces the result is equivalent to Oseledets multiplicative ergodic theorem.