Ergodic averages for commutative transformations along return times

TL;DR

This paper proves convergence of ergodic averages along specific return time sequences in mixing systems, extending prior results and addressing open questions about behavior along shrinking target sequences.

Contribution

It establishes both $L^2$ and pointwise convergence for ergodic averages along return time sequences generated by shrinking targets, for generic parameters and commuting transformations.

Findings

Proves convergence of ergodic averages along return time sequences.

Extends results to semi-random ergodic averages.

Addresses open questions in ergodic theory about shrinking targets.

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 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.