Extreme partitions of a Lebesgue space and their application in topological dynamics

Abstract

It is shown that any topological action $\Phi$ of a countable orderable and amenable group $G$ on a compact metric space $X$ and every $\Phi$-invariant probability Borel measure $\mu$ admit an extreme partition $\zeta$ of $X$ such that the equivalence relation $R_{\zeta}$ associated with $\zeta$ contains the asymptotic relation $A(\Phi)$ of $\Phi$. As an application of this result and the generalized Glasner theorem it is proved that $A(\Phi)$ is dense for the set $E_{\mu}(\Phi)$ of entropy pairs.