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arXiv · 1709.05373

A remark on the invertibility of semi-invertible cocycles

Lucas Backes
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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.

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Lucas Backes. 2017-09-15. A remark on the invertibility of semi-invertible cocycles. https://doi.org/10.1007/s10883-018-9420-0

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