Simple Lyapunov spectrum for certain linear cocycles over partially hyperbolic maps

TL;DR

This paper extends criteria for simple Lyapunov spectra from hyperbolic to partially hyperbolic systems, addressing new challenges and providing examples of stable simplicity in this broader context.

Contribution

It introduces new criteria for Lyapunov spectrum simplicity over partially hyperbolic maps, expanding the scope beyond hyperbolic systems and addressing associated technical issues.

Findings

Established criteria for simple Lyapunov spectra in partially hyperbolic skew-products.

Provided examples demonstrating stable simplicity in the partially hyperbolic setting.

Addressed recurrence of holonomy maps and disintegration issues in the new context.

Abstract

Criteria for the simplicity of the Lyapunov spectra of linear cocycles have been found by Furstenberg, Guivarc'h-Raugi, Gol'dsheid-Margulis and, more recently, Bonatti-Viana and Avila-Viana. In all the cases, the authors consider cocycles over hyperbolic systems, such as shift maps or Axiom A diffeomorphisms. In this paper we propose to extend such criteria to situations where the base map is just partially hyperbolic. This raises several new issues concerning, among others, the recurrence of the holonomy maps and the (lack of) continuity of the Rokhlin disintegrations of $u$-states. Our main results are stated for certain partially hyperbolic skew-products whose iterates have bounded derivatives along center leaves. They allow us, in particular, to exhibit non-trivial examples of stable simplicity in the partially hyperbolic setting.