# positive of the largest Lyapunov exponent

Let $$\alpha\in \mathbb{R} / \mathbb{Q}$$, $$$$A(x)=\left(\begin{array}{ll} \frac{1}{{\lambda}^2}-2 \cos 2\pi x -1& 2\lambda \cos 2\pi x-\frac{1}{{\lambda}} \\ \frac{1}{{\lambda}} & -1 \end{array}\right)$$$$ The Lyapunov exponent (LE) of $$(\alpha, A)$$ is given by $$$$LE(\lambda)=\lim _{n \rightarrow \infty} \frac{1}{n} \int_{\mathbb{R} / \mathbb{Z}} \ln \left\|A_{n}(x)\right\| d x$$$$ where $$A_{n}(x)=A(x+(n-1) \alpha) A(x+(n-2) \alpha) \cdots A(x).$$ How to prove there is a large positive number $$\lambda_0>0$$ so that when $$\lambda>\lambda_0,$$ the Lyapunov exponent is $$LE>0$$ ?

I think you can use the main result of the paper of Micheal Herman "Une méthode pour minorer..." https://link.springer.com/article/10.1007%2FBF02564647 in order to show that LE is positive when $$\lambda$$ is sufficiently large.