# The set of fiber-bunched cocycles with nonvanishing Lyapunov exponents over a partially hyperbolic map is open

## TL;DR

This paper proves that fiber-bunched SL(2,R)-valued cocycles with nonzero Lyapunov exponents form an open set over certain partially hyperbolic systems, highlighting the importance of accessibility in the base dynamics.

## Contribution

It establishes the openness of the set of fiber-bunched cocycles with nonvanishing Lyapunov exponents under specific conditions and provides a counterexample when accessibility is absent.

## Key findings

- Openness of fiber-bunched cocycles with nonzero Lyapunov exponents
- Counterexample showing necessity of accessibility
- Highlights role of accessibility in Lyapunov behavior

## Abstract

We prove that the set of fiber-bunched $SL(2,\mathbb{R})$-valued H\"{o}lder cocycles with nonvanishing Lyapunov exponents over a volume preserving, accessible and center-bunched partially hyperbolic diffeomorphism is open. Moreover, we present an example showing that this is no longer true if we do not assume acessibility in the base dynamics.