Exceptional ergodic directions in Eaton lenses

TL;DR

This paper constructs explicit examples of ergodic vertical flows in periodic Eaton lens configurations by analyzing infinite translation surfaces, showing that a significant set of lattices exhibit ergodic behavior with Hausdorff dimension exceeding 3/2.

Contribution

It provides explicit constructions of lattices with ergodic vertical flows in Eaton lens configurations and analyzes their Hausdorff dimension.

Findings

Hausdorff dimension of ergodic lattices > 3/2

Explicit construction of ergodic lattices

Analysis of infinite translation surfaces as covers of slit tori

Abstract

We construct examples of ergodic vertical flows in periodic configurations of Eaton lenses of fixed radius. We achieve this by studying a family of infinite translation surfaces that are $\mathbb{Z}^2$-covers of slit tori. We show that the Hausdorff dimension of lattices for which the vertical flow is ergodic is bigger than 3/2. Moreover, the lattices are explicitly constructed.