1 results
Weak ergodic averages over dilated measures
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 2 / February 2021
- Published online by Cambridge University Press:
- 07 October 2019, pp. 606-621
- Print publication:
- February 2021
-
- Article
- Export citation
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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 for
$$\begin{eqnarray}\lim _{t\in A,t\rightarrow \infty }\int _{\mathbb{R}^{m}}f(T_{t\unicode[STIX]{x1D6FC}}x)\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})=\int _{X}f\,d\unicode[STIX]{x1D707}\end{eqnarray}$$
$\unicode[STIX]{x1D707}$-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.
