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- W2977322277 abstract "Let $minmathbb{N}$ and $textbf{X}=(X,mathcal{X},mu,(T_{alpha})_{alphainmathbb{R}^{m}})$ 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 $Asubseteqmathbb{R}$ of density 1 such that for all $fin L^{infty}(mu)$, we have $$lim_{tin A,ttoinfty}int_{mathbb{R}^{m}}f(T_{t alpha}x),dnu(alpha)=int_{X}f,dmu$$ for $mu$-a.e. $xin X$. Let $W(textbf{X})$ denote the collection of all $alphainmathbb{R}^{m}$ such that the $mathbb{R}$-action $(T_{talpha})_{tinmathbb{R}}$ 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 $betainmathbb{R}^{m}$. 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." @default.
- W2977322277 created "2019-10-10" @default.
- W2977322277 creator A5047727947 @default.
- W2977322277 date "2019-10-07" @default.
- W2977322277 modified "2023-09-23" @default.
- W2977322277 title "Weak ergodic averages over dilated measures" @default.
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- W2977322277 doi "https://doi.org/10.1017/etds.2019.67" @default.
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