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arXiv · 2205.10135

Lipschitz sub-actions for locally maximal hyperbolic sets of a $C^1$ flow

Xifeng Su·Philippe Thieullen
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Abstract

Liv\v{s}ic theorem for flows asserts that a Lipschitz observable that has zero mean average along every periodic orbit is necessarily a coboundary, that is the Lie derivative of a Lipschitz function smooth along the flow direction. The positive Liv\v{s}ic theorem bounds from below the observable by such a coboundary as soon as the mean average along every periodic orbit is non negative. Previous proofs give a H\"older coboundary. Assuming that the dynamics is given by a locally maximal hyperbolic flow, we show that the coboundary can be Lipschitz. We introduce a new tool: the Lax-Oleinik semigroup, inspired by Fathi's weak KAM theory.

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Xifeng Su, Philippe Thieullen. 2022-05-20. Lipschitz sub-actions for locally maximal hyperbolic sets of a $C^1$ flow. https://arxiv.org/abs/2205.10135

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