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

Wojciech Bułatek, Brunon Kamiński, Jerzy Szymański

DOI: http://dx.doi.org/10.12775/TMNA.2019.007


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.


Topological $G$-action; relative entropy; entropy pair; asymptotic relation; relative Pinsker $\sigma$-algebra

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