A remark on the invertibility of semi-invertible cocycles

TL;DR

This paper shows that under certain conditions related to Lyapunov exponents, semi-invertible cocycles over hyperbolic systems are actually invertible, leading to a Livšic-type theorem for such cocycles.

Contribution

It establishes conditions under which semi-invertible cocycles become fully invertible and satisfy Livšic's theorem, extending the understanding of cocycle invertibility.

Findings

Semi-invertible cocycles are invertible under specific Lyapunov exponent conditions.

A Livšic-type theorem holds for certain semi-invertible cocycles.

Invertibility is guaranteed if the cocycle satisfies a product identity on fixed points.

Abstract

We observe that under certain conditions on the Lyapunov exponents a semi-invertible cocycle is, indeed, invertible. As a consequence, if a semi-invertible cocycle generated by a H\"{o}lder continuous map $A:M\to M(d, \mathbb{R})$ over a hyperbolic map $f:M\to M$ satisfies a Liv\v{s}ic's type condition, that is, if $A(f^{n-1}(p))\cdot\ldots \cdot A(f(p))A(p)=\text{Id}$ for every $p\in \text{Fix}(f^n)$ then the cocycle is invertible, meaning that $A(x)\in GL(d,\mathbb{R})$ for every $x\in M$, and a Liv\v{s}ic's type theorem is satisfied.