Nonexpanding maps, Busemann functions, and multiplicative ergodic theory

We describe some results on the dynamics of nonexpanding maps of metric spaces. First, considering nonexpanding maps of proper metric spaces, we explain generalizations of some results of Bearden, which extends the Wolff-Denjoy theorem in complex analysis. Second, we consider certain cocycles, or 'random products', of nonexpanding maps of nonpositively curved spaces. In a joint work with Maxgulis, we obtained that almost every trajectory lies on sublinear distance from a geodesic ray. In the special case where the metric space is the symmetric space of positive definite, symmetric matrices and the cocycles are isometries, this statement is equivalent to Oseledec's multiplicative ergodic theorem. Further consequences concerning random ergodic theorems, cocycles of bounded linear operators, random walks and Poisson boundaries, are briefly discussed.