Gowers' Ramsey Theorem with multiple operations and dynamics of the homeomorphism group of the Lelek fan

TL;DR

This paper extends Gowers' Ramsey theorem to multiple operations and demonstrates that the homeomorphism group of the Lelek fan, preserving a typical linear order, is extremely amenable, revealing deep symmetry properties.

Contribution

It generalizes Gowers' Ramsey theorem to multiple operations and applies it to prove the extreme amenability of a symmetry-preserving homeomorphism group of the Lelek fan.

Findings

Generalized Gowers' Ramsey theorem for multiple operations

Proved the extreme amenability of the homeomorphism group of the Lelek fan

Identified symmetry properties of the Lelek fan's homeomorphism group

Abstract

We generalize the finite version of Gowers' Ramsey theorem to multiple tetris-like operations and apply it to show that a group of homeomorphisms that preserve a "typical" linear order of branches of the Lelek fan, a compact connected metric space with many symmetries, is extremely amenable.