Characterization of uniform hyperbolicity for fiber-bunched cocycles

Let $(T;A)$ be a $SL(2;\mathbb{R})$-cocycle defined over a transitive subshift of finite type. We will show that the cocycle $(T;A)$ is uniformly hyperbolic if the cocycle satisfies the fiber-bunching condition, and there is a constant $\tau > 0$ and a full probability set $\mathcal{S}\subset \mathbb{R}$ such that $\lambda_{+}(x)\geq \tau$ for every $x \in \mathcal{S}$.

The talk follows the proofs of R.Velozo Ruiz MSc dissertation, based on C. Bonatti, X. Gómez-Mont, M. Viana's paper and M. Viana's book.