On ergodicity of foliations on $\mathbb{Z}^d$-covers of half-translation surfaces and some applications to periodic systems of Eaton lenses

TL;DR

This paper investigates the ergodic properties of geodesic flows in periodic Eaton lens patterns, establishing ergodicity for certain $ obreak bZ^d$-coverings of quadratic differentials with zero Lyapunov exponents, with applications to optical systems.

Contribution

It proves ergodicity of geodesic flows on $bZ^d$-covers of quadratic differentials, linking geometric dynamics with optical lens configurations.

Findings

Identifies ergodic geodesic flows in Eaton lens patterns.

Establishes a connection between quadratic differentials and optical systems.

Proves ergodicity for $bZ^d$-covers with vanishing Lyapunov exponents.

Abstract

We consider the geodesic flow defined by periodic Eaton lens patterns in the plane and discover ergodic ones among those. The ergodicity result on Eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori. Ergodicity itself is concluded for $\mathbb{Z}^d$-covers of quadratic differentials on compact surfaces with vanishing Lyapunov exponents.