A remark on the invertibility of semi-invertible cocycles

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 Hölder continuous map $A:M\to M(d,\mathbb{R})$ over a hyperbolic map $f:M\to M$ satisfies a Livšic's type condition, that is, if $A(f^{n-1}(p))\cdot \dots \cdot A(f(p))A(p)=\text{Id}$ 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 Livšic's type theorem is satisfied.

Some corrections to the previous version were made

References in corpus (2)

• Nonexistence of Lyapunov Exponents for Matrix Cocycles
• On the periodic approximation of Lyapunov exponents for semi-invertible cocycles