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arXiv · 2010.08945

Historic behaviour vs. physical measures for irrational flows with multiple stopping points

Martin Andersson·Pierre-Antoine Guihéneuf
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Abstract

We study Birkhoff averages along trajectories of smooth reparameterizations of irrational linear flows of the two torus with two stopping points, say $\mathbf p$ and $\mathbf q$, of quadratic order. The limiting behaviour of such averages is independent of the starting point in a set of full Haar-Lebesgue measure and depends in an intricate way on the Diophantine properties of both the slope $\alpha$ of the linear flow as well as the relative position of $\mathbf p$ and $\mathbf q$. In particular, if $\alpha$ is Diophantine, then Birkhoff limits diverge almost everywhere (historic behaviour) and if $\alpha$ is sufficiently Liouville, then there exists some $\mathbf p$ and $\mathbf q$ such that the Birkhoff averages converge almost everywhere (unique physical measure).

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Martin Andersson, Pierre-Antoine Guihéneuf. 2020-10-18. Historic behaviour vs. physical measures for irrational flows with multiple stopping points. https://arxiv.org/abs/2010.08945

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