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arXiv · 1408.5639

Liv\v{s}ic Theorem for Banach Rings

Genady Ya. Grabarnik·Misha Guysinsky
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Abstract

We prove the Liv\v{s}ic Theorem for H\"{o}lder continuous cocycles with values in Banach rings. We consider a transitive homeomorphism ${\sigma:X\to X}$ that satisfies the Anosov Closing Lemma, and a H\"{o}lder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. We show that it is a coboundary with a H\"{o}lder continuous transition function if and only if ${a(\sigma^{n-1}p)\ldots a(\sigma p)a(p)=e}$ for each periodic point $p=\sigma^n p$.

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Genady Ya. Grabarnik, Misha Guysinsky. 2014-08-24. Liv\v{s}ic Theorem for Banach Rings. https://arxiv.org/abs/1408.5639

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