Lyapunov spectrum via boundary theory I
Uri Bader, Alex Furman
TL;DR
The article develops a boundary-theoretic framework for analyzing Lyapunov spectra of measurable cocycles valued in connected semisimple real Lie groups over ergodic probability-preserving systems. It proves that for integrable Zariski-dense representations within Apafic Gregs, the Lyapunov spectrum is simple, and it depends continuously on uniform integrability data, with positive drift when the image is non-solvable-by-compact. The approach unifies and extends classical results for random walks (Furstenberg, Guivarc'h–Raugi, Gol'dsheid–Margulis, Le Page) and applies to geometric contexts such as geodesic flows in negative curvature, Gibbs measures, and representations of fundamental groups, via AREA and boundary theory. Overall, the paper provides a general, soft-framework for Lyapunov theory that yields new perspectives and broad applicability, paving the way for further examples and refinements in a companion work.
Abstract
This paper is concerned with the Lyapunov spectrum for measurable cocycles over an ergodic pmp system taking values in semi-simple real Lie groups. We prove simplicity of the Lyapunov spectrum and its continuity under certain perturbations for a class systems that includes many familiar examples. Our framework uses some soft qualitative assumptions, and does not rely on symbolic dynamics. We use ideas from boundary theory that appear in the study of super-rigidity to deduce our results. This gives a new perspective even on the most studied case of random matrix products. The current paper introduces the general framework and contains the proofs of the main results and some basic examples. In a follow up paper we discuss further examples.