Very recently Adam Bartoš, Tristan Bice and Alessandro Vignati discovered a duality, generalizing the Stone duality, between second countable \(T_1\) compacta and \(\omega\)-posets. Their approach allows for elementary combinatorial constructions, in the spirit of Fraïssé theory, of classical continua such as the Lelek fan or the pseudo-arc.
We extend this framework to study homeomorphism groups of compacta. We characterize Hausdorff compacta such that their group of homeomorphisms has a dense or a comeager conjugacy class. We use this characterization to prove that there exists a comeager conjugacy class in the group of homeomorphisms of the Lelek fan. This sheds light on the dynamics on the Lelek fan: a generic homeomorphism has no Lie-Yorke pair; in particular, its topological entropy is zero. We also show that there is a homeomorphism of the pseudo-arc with a dense conjugacy class.
This is joint work with Tristan Bice.