## Topological Methods in Nonlinear Analysis

### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 53, Number 2 (2019), 447-455.

Dates
First available in Project Euclid: 9 May 2019

https://projecteuclid.org/euclid.tmna/1557367217

Digital Object Identifier
doi:10.12775/TMNA.2019.007

Mathematical Reviews number (MathSciNet)
MR3983981

#### Citation

Bułatek, Wojciech; Kamiński, Brunon; Szymański, Jerzy. Extreme partitions of a Lebesgue space and their application in topological dynamics. Topol. Methods Nonlinear Anal. 53 (2019), no. 2, 447--455. doi:10.12775/TMNA.2019.007. https://projecteuclid.org/euclid.tmna/1557367217

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