Variation-norm and fluctuation estimates for ergodic bilinear averages

TL;DR

This paper establishes new variation-norm and fluctuation estimates for ergodic bilinear averages, demonstrating polynomial growth bounds on fluctuations and extending bilinear Lp bounds through advanced inequalities.

Contribution

It introduces novel variation-norm bounds for ergodic bilinear averages and extends maximal Bessel inequalities to analyze fluctuations.

Findings

Fluctuations grow at most polynomially with scale.

Almost sure convergence of bilinear averages is strengthened.

New bilinear Lp bounds for variation-norms are established.

Abstract

For any dynamical system, we show that higher variation-norms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respect to (the growth of) the underlying scale. These results strengthen previous works of Lacey and Bourgain where almost surely convergence of the sequence was proved (which is equivalent to the qualitative statement that the number of fluctuations is finite at each scale). Via transference, the proof reduces to establishing new bilinear Lp bounds for variation-norms of truncated bilinear operators on R, and the main ingredient of the proof of these bounds is a variation-norm extension of maximal Bessel inequalities of Lacey and Demeter--Tao--Thiele.