El Hadji Yaya Tall | Searcharxiv

arXiv2026

Log-Höder continuity at zero Lyapunov gap for finite state Markov $GL(2)$-cocycles

We prove that the extremal Lyapunov exponents of finite-state Markov $\mathrm{GL}(2,\mathbb{R})$-cocycles are pointwise log-Hölder continuous, jointly in the cocycle matrices and the transition kernel, at every parameter $(A,P)$ satisfying $λ_+(A,P)=λ_-(A,P)$. Perturbations are taken within a fixed transition graph. The main new ingredient is a Markov perpetuity estimate for the nonsplit triangular case, obtained through a martingale--coboundary decomposition. In the conformal case, the exponent $1/2$ in the logarithmic modulus can be replaced by $1$.

math.DS↗

arXiv2021

Analyticity of the Lyapunov exponents of random products of quasi-periodic cocycles

We show that the top Lyapunov exponent $λ_+(p)$ , $p = (p_1, \cdots, p_N)$ with $p_i >0$ for each $i$, associated with a random product of quasi-periodic cocycles depends real analytically on the transition probabilities $p$ whenever $λ_+(p)$ is simple. Moreover if the spectrum at $p$ is simple (all Lyapunov exponents having multiplicity one ) then all Lyapunov exponents depend real analytically on $p$.

math.DS↗