Weak ergodic averages over dilated measures

Abstract

Let $m\in \mathbb{N}$ and $\mathbf{X}=(X,{\mathcal{X}},\unicode[STIX]{x1D707},(T_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}})$ be a measure-preserving system with an $\mathbb{R}^{m}$-action. We say that a Borel measure $\unicode[STIX]{x1D708}$ on $\mathbb{R}^{m}$ is weakly equidistributed for $\mathbf{X}$ if there exists $A\subseteq \mathbb{R}$ of density 1 such that, for all $f\in L^{\infty }(\unicode[STIX]{x1D707})$, we have $$\begin{eqnarray}\lim _{t\in A,t\rightarrow \infty }\int _{\mathbb{R}^{m}}f(T_{t\unicode[STIX]{x1D6FC}}x)\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})=\int _{X}f\,d\unicode[STIX]{x1D707}\end{eqnarray}$$ for $\unicode[STIX]{x1D707}$-almost every $x\in X$. Let $W(\mathbf{X})$ denote the collection of all $\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}$ such that the $\mathbb{R}$-action $(T_{t\unicode[STIX]{x1D6FC}})_{t\in \mathbb{R}}$ is not ergodic. Under the assumption of the pointwise convergence of the double Birkhoff ergodic average, we show that a Borel measure $\unicode[STIX]{x1D708}$ on $\mathbb{R}^{m}$ is weakly equidistributed for an ergodic system $\mathbf{X}$ if and only if $\unicode[STIX]{x1D708}(W(\mathbf{X})+\unicode[STIX]{x1D6FD})=0$ for every $\unicode[STIX]{x1D6FD}\in \mathbb{R}^{m}$. Under the same assumption, we also show that $\unicode[STIX]{x1D708}$ is weakly equidistributed for all ergodic measure-preserving systems with $\mathbb{R}^{m}$-actions if and only if $\unicode[STIX]{x1D708}(\ell )=0$ 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.