Homeomorphisms of continua through projective Fra\"iss\'e limits

TL;DR

This paper explores the structure of homeomorphism groups of the pseudoarc using projective Fra"iss"e limits, revealing dense conjugacy orbits and the existence of non-conjugate automorphisms.

Contribution

It introduces a novel approach linking homeomorphism groups of continua to automorphism groups of projective Fra"iss"e limits, extending previous results.

Findings

Dense orbit of the diagonal conjugacy action on homeomorphism groups

Existence of a homeomorphism not conjugate to any automorphism of the pre-pseudoarc

Strengthening of existing results on homeomorphism groups of the pseudoarc

Abstract

We study homeomorphisms and the homeomorphism groups of compact metric spaces using the automorphism groups of projective Fra\"iss\'e limits. In our applications, we investigate the Polish group ${\rm Homeo}(P)$ of all homeomorphisms of the pseudoarc $P$ using the automorphism group ${\rm Aut}(\mathbb{P})$ of the pre-pseudoarc $\mathbb{P}$. Strengthening results from the literature, we show that the diagonal conjugacy action of ${\rm Homeo}(P)$ on ${\rm Homeo}(P)^{\mathbb{N}}$ has a dense orbit. In our second application, we show that there exists a homeomorphism of $P$ that is not conjugate in ${\rm Homeo}(P)$ to an element of ${\rm Aut}(\mathbb{P})$.