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arXiv · 2112.06131

Ergodic Deviations of Degenerate Multidimensional Actions -- Symmetric Convex Bodies

Hao Wu
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Abstract

We prove that the ergodic deviation of a degenerate $\mathbb{Z}^2$-action on the torus $\mathbb{T}^2$ relative to a symmetric, strictly convex body can be decomposed into two parts, and that each part admits a limit distribution after choosing a suitable normalizer. Specifically, the first part is similar to an ergodic sum of smooth observables after being normalized by $N$, and the second part is similar to the case of a random toral translation, i.e., the $\mathbb{Z}$-action, but with a normalizer of $N^{\frac{1}{2}}$. The key difference is that we employ the product flow on the product space of $\mathbb{Z}^2$ lattices for the multidimensional action.

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Hao Wu. 2021-12-12. Ergodic Deviations of Degenerate Multidimensional Actions -- Symmetric Convex Bodies. https://arxiv.org/abs/2112.06131

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