A Livšic Theorem for Matrix Cocycles Over Non-uniformly Hyperbolic Systems
Tóm tắt
We prove a Livšic-type theorem for Hölder continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever $$(f,\mu )$$ is a non-uniformly hyperbolic system and $$A:M \rightarrow GL(d,\mathbb {R}) $$ is an $$\alpha $$-Hölder continuous map satisfying $$ A(f^{n-1}(p))\ldots A(p)=\text {Id}$$ for every $$p\in \text {Fix}(f^n)$$ and $$n\in \mathbb {N}$$, there exists a measurable map $$P:M\rightarrow GL(d,\mathbb {R})$$ satisfying $$A(x)=P(f(x))P(x)^{-1}$$ for $$\mu $$-almost every $$x\in M$$. Moreover, we prove that whenever the measure $$\mu $$ has local product structure the transfer map P is $$\alpha $$-Hölder continuous in sets with arbitrary large measure.
Tài liệu tham khảo
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arXiv:1711.02135
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