Distal measures for semigroup actions
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초록
We introduce the notion of distal measure of a semigroup action $\phi$ on a compact metric space $X$, and show that the set of distal measures of $\phi$ is a $G_{\delta \sigma}$ subset of the space $\mathcal M(X)$ of Borel probability measures on $X$ with the weak* topology. Moreover, we prove that the distal measures of $\phi$ with full support is dense in $\mathcal M(X)$ under a suitable condition.
Abstract
We introduce the notion of distal measure of a semigroup action $\phi$ on a compact metric space $X$, and show that the set of distal measures of $\phi$ is a $G_{\delta \sigma}$ subset of the space $\mathcal M(X)$ of Borel probability measures on $X$ with the weak* topology. Moreover, we prove that the distal measures of $\phi$ with full support is dense in $\mathcal M(X)$ under a suitable condition.