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

TL;DR

This paper proves that under certain semi-continuity conditions of Oseledets flags in a dynamical system, one can construct Pesin sets with exponentially small tails, linking geometric and measure-theoretic properties.

Contribution

It establishes a connection between the semi-continuity of Oseledets flags and the existence of Pesin sets with exponentially small tails in the context of linear cocycles over subshifts.

Findings

Oseledets flags depend upper semi-continuously on the base point.

Existence of Pesin sets with exponentially small tails under semi-continuity conditions.

Application to invertible linear cocycles over subshifts of finite type.

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 H\"older continuous potential on $X$, and let $A$ be an invertible continuous linear cocycle over $f$ acting on a continuous $\R^d$-bundle $E$ over $X$ with Lyapunov exponents $\lambda_k < \lambda_{k-1} < \ldots < \lambda_1$ 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 $\mu$.