Ergodic averages for commutative transformations along return times
Sebastián Donoso, Sovanlal Mondal, Vicente Saavedra-Araya
TL;DR
The paper studies ergodic averages along return-time sequences generated by shrinking targets, proving $L^2$ and, in some cases, pointwise convergence for single and multiple averages along the sequences $a_n(y)$ with $a olinebreak[4] ext{ in }(0,1)$. It develops a deterministic–random bridge via the lacunary trick and dyadic interval approximations to reduce to independent blocks, and extends these results to semi-random settings under decay of correlations. The authors establish A–D type results: $L^2$ convergence for single and multiple averages along return times (Theorems A and B), a semi-random ergodic theorem under strong mixing (Theorem C), and a semi-random two-term averaging result (Theorem D), with pointwise convergence in many commuting-power cases. The work advances understanding of how shrinking targets and return times interact with ergodic averages, linking deterministic dynamics to random-like convergence behavior even in dependent settings.
Abstract
In this paper, we extend recent results on the convergence of ergodic averages along sequences generated by return times to shrinking targets in rapidly mixing systems, partially answering questions posed by the first author, Maass and the third author in [6]. In particular, for a fixed parameter $a\in (0,1)$ and for generic $y\in [0,1]$, we establish both $L^2$ and pointwise convergence for single averages and multiple averages for commuting transformations along the sequences $(a_n(y))_{n\in \mathbb{N}}$, obtained by arranging the set $$\Big\{n\in\mathbb{N}: 0<2^ny \mod{1}<n^{-a} \Big\}$$ in an increasing order. We also obtain new results for semi-random ergodic averages along sequences of similar type.