SUBADDITIVE COCYCLES AND HOROFUNCTIONS

Abstract

Subadditive cocycles are the random version of subadditive sequences. They play an important role in probability and ergodic theory, notably through Kingman’s theorem ensuring their almost sure convergence. We discuss a variation around Kingman’s theorem, showing that a subadditive cocycle is in fact almost additive at many times. This result is motivated by the study of the iterates of deterministic or random semicontractions on metric spaces, and implies the almost sure existence of a horofunction determining the behavior at infinity of such a sequence. In turn, convergence at infinity follows when the geometry of the space has some features of nonpositive curvature. The aim of this text is to present and put in perspective the results we have proved with Anders Karlsson in the article Gouëzel and Karlsson [2015]. The topic of this article is the study, in an ergodic theoretic context, of some subadditivity properties, and their relationships with dynamical questions with a more geometric flavor, dealing with the asymptotic behavior of random semicontractions on general metric spaces. This text is translated from an article written in French on the occasion of the first congress of the French Mathematical Society Gouëzel, Sébastien [2017]. The proof of the main ergodic-theoretic result in Gouëzel and Karlsson [2015] has been completely formalized and checked in the computer proof assistant Isabelle/HOL Gouëzel, Sébastien [2016]. 1 Iteration of a semicontraction on Euclidean space In order to explain the problems we want to consider, it is enlightening to start with a more elementary example, showing how subadditivity techniques can be useful to understand a deterministic semicontraction. In the next section, we will see how these results can be extended to random semicontractions. MSC2010: primary 37H15; secondary 37A30.