Weak ergodic averages over dilated measures

TL;DR

This paper characterizes weak equidistribution of measures over dilated measures in ergodic systems, linking it to the measure's nullity on certain sets and hyperplanes, using ergodic theory instead of harmonic analysis.

Contribution

It provides a novel ergodic theoretic characterization of weak equidistribution for measures on based on the structure of non-ergodic directions and hyperplanes, under pointwise convergence assumptions.

Findings

Weakly equidistributed measures vanish on non-ergodic directions.

Measures equidistributed for all ergodic systems are supported outside hyperplanes.

Results are obtained without harmonic analysis, relying solely on ergodic theory.

Abstract

Let $m\in\mathbb{N}$ and $\textbf{X}=(X,\mathcal{X},\mu,(T_{\alpha})_{\alpha\in\mathbb{R}^{m}})$ be a measure preserving system with an $\mathbb{R}^{m}$-action. We say that a Borel measure $\nu$ on $\mathbb{R}^{m}$ is weakly equidistributed for $\textbf{X}$ if there exists $A\subseteq\mathbb{R}$ of density 1 such that for all $f\in L^{\infty}(\mu)$, we have $$\lim_{t\in A,t\to\infty}\int_{\mathbb{R}^{m}}f(T_{t \alpha}x)\,d\nu(\alpha)=\int_{X}f\,d\mu$$ for $\mu$-a.e. $x\in X$. Let $W(\textbf{X})$ denote the collection of all $\alpha\in\mathbb{R}^{m}$ such that the $\mathbb{R}$-action $(T_{t\alpha})_{t\in\mathbb{R}}$ is not ergodic. Under the assumption of the pointwise convergence of double Birkhoff ergodic average, we show that a Borel measure $\nu$ on $\mathbb{R}^{m}$ is weakly equidistributed for an ergodic system $\textbf{X}$ if and only if $\nu(W(\textbf{X})+\beta)=0$ for every $\beta\in\mathbb{R}^{m}$. Under the same assumption, we also show that $\nu$ is weakly equidistributed for all ergodic measure preserving systems with $\mathbb{R}^{m}$-actions if and only if $\nu(\ell)=0$ for all hyperplanes $\ell$ of $\mathbb{R}^{m}$. Unlike many equidistribution results in literature whose proofs use methods from harmonic analysis, our results adopt a purely ergodic theoretic approach.