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)