A Remark on the Invertibility of Semi-invertible Cocycles
Tóm tắt
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
$A:\mathcal {M}\to M(d, \mathbb {R})$
over a hyperbolic system
$f:\mathcal {M}\to \mathcal {M}$
satisfies a Livšic’s type condition, that is, if A(fn− 1(p)) ⋅… ⋅ A(f(p))A(p) = Id for every p ∈ Fix(fn), then the cocycle is invertible, meaning that
$A(x)\in GL(d,\mathbb {R})$
for every
$x\in \mathcal {M}$
, and a Livšic’s type theorem is satisfied.
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