Universal minimal flow in the language of near filters and its applications

TL;DR

This paper introduces a novel framework using near ultrafilters to analyze the greatest ambit and universal minimal flow, simplifying proofs and providing new results in topological dynamics and group amenability.

Contribution

It develops a unified language of near ultrafilters for topological dynamics, offering simpler proofs and new insights into the amenability of isometry groups.

Findings

Groups of isometries of generalized Urysohn spaces are extremely amenable.

Simplified proofs of known results in topological dynamics.

Partial resolution of Pestov's conjecture on the Ellis problem.

Abstract

We describe the greatest ambit and the universal minimal flow as spaces of near ultrafilters. We translate other notions of topological dynamics into this language and show how this approach simplifies some known proofs. We provide a simple proof that groups of isometries of generalized Urysohn spaces are extremely amenable without use of concentration of measure phenomena and we give a partial answer to Pestov's conjecture to the problem of Ellis.