Norm convergence of continuous-time polynomial multiple ergodic averages

TL;DR

This paper proves the $L^2$-norm convergence of continuous-time polynomial multiple ergodic averages for measure-preserving actions of $ ^D$, confirming a continuous analogue of a long-standing conjecture in ergodic theory.

Contribution

It introduces a new inductive scheme using fractional powers of time variables, providing an alternative to PET induction for establishing convergence.

Findings

Proves $L^2$ convergence of continuous-time polynomial averages.

Develops a new inductive method based on fractional time scaling.

Confirms the continuous-time analogue of a major open conjecture.

Abstract

For a jointly measurable probability-preserving action $\tau:\mathbb{R}^D\curvearrowright (X,\mu)$ and a tuple of polynomial maps $p_i:\mathbb{R}\to \mathbb{R}^D$, $i=1,2,...,k$, the multiple ergodic averages \[ \frac{1}{T}\int_0^T (f_1\circ \tau^{p_1(t)})(f_2\circ \tau^{p_2(t)})... (f_k\circ \tau^{p_k(t)})\,\mathrm{d} t \] converge in $L^2(\mu)$ as $T \to \infty$ for any $f_1,f_2,...,f_k \in L^\infty(\mu)$. This confirms the continuous-time analog of the conjectured norm convergence of discrete polynomial multiple ergodic averages, which in is its original formulation remains open in most cases. A proof of convergence can be given based on the idea of passing up to a sated extension of $(X,\mu,\tau)$ in order to find simple characteristic factors, similarly to the recent development of this idea for the study of related discrete-time averages, together with a new inductive scheme on tuples of polynomials. The new induction scheme becomes available upon changing the time variable in the above integral by some fractional power, and provides an alternative to Bergelson's PET induction, which has been the mainstay of positive results in this area in the past.