Ergodic Deviations of Degenerate Multidimensional Actions -- Symmetric Convex Bodies

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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