## – Europe/Lisbon

Room P3.10, Mathematics Building — Online

Hölder Continuity of Lyapunov Exponents in Random Products of Invertible Matrices

Consider a probability measure $\nu$ with compact support in $GL(d)$ and a linear cocycle $\mathcal{F}:GL^{\mathbb{N}}(d)\times {\mathbb{R}}^d\rightarrow GL^{\mathbb{N}}(d)\times {\mathbb{R}}^d$ defined as $\mathcal{F}((g_j)_j,v)=((g_{j+1})_j,g_0 v)$. The cocycle associated with a random product of $2\times 2$ invertible matrices (i.e., $d=2$) under a probability distribution $\nu$ have two (possibly equal) Lyapunov exponents $\lambda_1(\nu)\ge\lambda_2(\nu)$. When $\lambda_1>\lambda_2$ we can prove that those two exponents are pointwise Hölder continuous with respect to the probability measure $\nu$. A natural question arises: does this result generalize to higher dimensions? In this talk, we'll explore the concept of a stochastic dynamical system, define a random $GL(d)$-cocycle and investigate the above-mentioned generalization. This is a work in progress with Adriana Sánchez, El Hadji Yaya Tall and Marcelo Viana.