On asymptotic expansions of ergodic integrals for $\mathbb Z^d$-extensions of translation flows
Volume 284 / 2025
Studia Mathematica 284 (2025), 229-280
MSC: Primary 37E35; Secondary 28D05, 37H05, 14H30, 60F15
DOI: 10.4064/sm240425-15-4
Published online: 30 September 2025
Abstract
We obtain expansions of ergodic integrals for $\mathbb Z^d$-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 ($\mathbb Z$-extensions of self-similar translation flows of genus-$2$ surfaces), and particular cases of the Ehrenfest wind-tree model.
