Weak ergodic averages over dilated measures

Abstract

Let $m\in \mathbb{N}$ and $\text{X}=(X,\mathcal{X},\mu ,(T_{\alpha }{)}_{\alpha \in {\mathbb{R}}^m})$ 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 $\text{X}$ if there exists $A\subseteq \mathbb{R}$ of density 1 such that for all $f\in L^{\infty }(\mu )$, we have $$\underset{t\in A,t\to \infty }{\lim }{\int }_{{\mathbb{R}}^m}f(T_{t\alpha }x)d\nu (\alpha )={\int }_{X}fd\mu$$ for $\mu$-a.e. $x\in X$. Let $W(\text{X})$ 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 $\text{X}$ if and only if $\nu (W(\text{X})+\beta )=0$ 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.

18 pages