A positive proportion Livshits theorem

Caleb Dilsavor; James Marshall Reber

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arXiv · 2304.01372

A positive proportion Livshits theorem

Caleb Dilsavor·James Marshall Reber

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Abstract

Given a transitive Anosov diffeomorphism or flow on a closed connected Riemannian manifold $M$, the Livshits theorem states that a H\"{o}lder function $\varphi : M \to \mathbb{R}$ is a coboundary if all of its periods vanish. We explain how a finer statistical understanding of the distribution of these periods can be used to obtain a stronger version of the classical Livshits theorem where one only has to check that the periods of $\varphi$ vanish on a set of positive asymptotic upper density. We also include a strengthening of the nonpositive Livshits theorem.

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Caleb Dilsavor, James Marshall Reber. 2023-04-03. A positive proportion Livshits theorem. https://arxiv.org/abs/2304.01372

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