Semi-continuity of Oseledets flags and Pesin sets with exponentially small tails

Abstract

Let $f$ be an invertible transitive subshift of finite type over a bilateral symbol space $X$, let $\mu$ be a Gibbs measure for $f$ determined by a Hölder continuous potential on $X$, and let $A$ be an invertible continuous linear cocycle over $f$ acting on a continuous ${\mathbb{R}}^d$-bundle $E$ over $X$ with Lyapunov exponents ${\lambda }_k<{\lambda }_{k-1}<\dots <{\lambda }_1$ such that $A^{-1}$ is continuous as well. We prove that if the Oseledets flags $F_j(x)=E_j(x)\oplus E_{j-1}(x)\oplus \cdots \oplus E_1(x)$ depend upper semi-continuously on $x\in X$, then there exists a Pesin set with exponentially small tails for $\mu$.