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

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.