This analysis reveals expansions of ergodic integrals and weak rational ergodicity in Z-extensions, suggesting optimal rates.
We obtain expansions of ergodic integrals for Zd-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 translation flows of genus-2 surfaces), and particular cases of the Ehrenfest wind-tree model.
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Bruin et al. (2025) studied this question.
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