Livšic Theorem for matrix cocycles

Abstract

We prove the Livšic Theorem for arbitrary $\mathrm{GL}(m,\Bbb{R})$ cocycles. We consider a hyperbolic dynamical system $f : X \to X$ and a Hölder continuous function $A: X \to \mathrm{GL}(m,\Bbb{R})$. We show that if $A$ has trivial periodic data, i.e. $A(f^{n-1} p) \cdots A(fp) A(p)$ $= \mathrm{Id}$ for each periodic point $p=f^n p$, then there exists a Hölder continuous function $C: X \to \mathrm{GL}(m,\Bbb{R})$ satisfying $A (x) = C(f x) C(x) ^{-1}$ for all $x \in X$. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on growth of products along orbits for an arbitrary Hölder function $A$.