On asymptotic expansions of ergodic integrals for $\Z^d$-extensions of translation flows

TL;DR

This paper derives asymptotic expansions for ergodic integrals in certain $ ext{Z}^d$-extensions of translation flows, revealing weak rational ergodicity with optimal rates, and applies results to specific models like staircase flows and wind-tree models.

Contribution

It provides new asymptotic expansions for ergodic integrals in $ ext{Z}^d$-extensions of translation flows, advancing understanding of their ergodic properties.

Findings

Established asymptotic expansions for ergodic integrals.

Demonstrated weak rational ergodicity with optimal rates.

Applied results to staircase flows and wind-tree models.

Abstract

We obtain expansions of ergodic integrals for $\Z^d$-covers of compact self-similar translation flows, and as a consequence we obtain a form of weak rational ergodicity with optimal rates. As examples, we consider the so-called self-similar $(s,1)$-staircase flows ($\Z$-extensions of self-similar translations flows of genus-$2$ surfaces), and particular cases of the Ehrenfest wind-tree model.