# Random product of quasi-periodic cocycles

## TL;DR

This paper investigates the stability and generic properties of Lyapunov exponents for random products of quasi-periodic cocycles, showing openness, density, and continuity results in various matrix group settings.

## Contribution

It establishes that the set of cocycles with positive Lyapunov exponent is open and dense, and that Lyapunov exponents are continuous at many points, extending results to higher dimensions and specific structures.

## Key findings

- Positive Lyapunov exponent set is open and dense in certain cocycle spaces.
- Lyapunov exponents are continuous at a dense set of points.
- Results extend to higher-dimensional cocycles with diagonal structures.

## Abstract

Given a finite set of quasi-periodic cocycles the random product of them is defined as the random composition according to some probability measure. We prove that the set of $C^r$, $0\leq r \leq \infty$ (or analytic) $k+1$-tuples of quasi periodic cocycles taking values in $SL_2(\mathbb{R})$ such that the random product of them has positive Lyapunov exponent contains a $C^0$ open and $C^r$ dense subset which is formed by $C^0$ continuity point of the Lyapunov exponent For $k+1$-tuples of quasi periodic cocycles taking values in $GL_d(\mathbb{R})$ for $d>2$, we prove that if one of them is diagonal, then there exists a $C^r$ dense set of such $k+1$-tuples which has simples Lyapunov spectrum and are $C^0$ continuity point of the Lyapunov exponent.