Ergodic Deviations of Degenerate Multidimensional Actions -- Symmetric Convex Bodies

TL;DR

This paper analyzes the ergodic deviation of a degenerate multidimensional action on the torus relative to symmetric convex bodies, revealing a decomposition into two parts each with a limit distribution under suitable normalization.

Contribution

It introduces a novel decomposition of ergodic deviations for degenerate $bZ^2$-actions on the torus and establishes limit distributions for each part using product flow techniques.

Findings

Decomposition of ergodic deviation into two parts with distinct limit behaviors.

Identification of normalizers $N$ and $N^{1/2}$ for the two parts.

Application of product flow on $bZ^2$ lattices for analysis.

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.