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.