The study of random products of operators appears naturally in many

areas of mathematics and its applications. An example of this is product

of isometries of some metric space. Much like the Oseledets’ theorem

governs the behaviour of products of linear operators, the metric setting

can be described with the multiplicative ergodic theorem of Karlsson and

Gouezel [1].

In this talk we will focus on the specific case where the metric space

is a strongly hyperbolic. These spaces exhibit very nice properties with

respect to their action, which will allow us to obtain a more descriptive

ergodic theorem as well as some regularity results for the process on the

space driven by the action of the successive isometries.

This work is the result of my PhD thesis

[1] Gouezel, S., Karlsson, A. “Subadditive and multiplicative ergodic theorems,” in the jour-

nal, J. Eur. Math. Soc. (JEMS) 22, 1893-1915 (2020)

[2] Sampaio, L.M. “Continuity of the drift in groups acting on strongly hyperbolic spaces” in

the arxiv: https://arxiv.org/abs/2204.08299, (preprint)