Homeomorphisms of continua through projective Fraïssé limits
Márk Poór, Sławomir Solecki
TL;DR
This work develops a projective Fraïssé framework to relate automorphism groups of Fraïssé limits to the homeomorphism groups of their canonical quotients, focusing on the pseudoarc. By analyzing topologies on ${\rm Aut}({\mathbb K})$ induced from ${\rm Homeo}(K)$ and introducing a Joint Projection Property scheme, the authors translate combinatorial density properties into dynamical statements about diagonal conjugacy actions. In the pseudoarc case, they prove that for every $n$, the diagonal conjugacy action of ${\rm Aut}({\mathbb P})$ on ${\rm Aut}({\mathbb P})^n$ is dense in ${\rm Homeo}(P)^n$ and that these orbits are comeager, yielding strong structural information about ${\rm Homeo}(P)$. Moreover, they construct a homeomorphism of the pseudoarc not conjugate to any automorphism of the pre-pseudoarc, establishing a sharp limitation on the extent to which ${\rm Homeo}(P)$ can be generated by ${\rm Aut}({\mathbb P})$-type symmetries. Overall, the paper provides a robust bridge between projective Fraïssé theory and the dynamics of continuum homeomorphism groups, with concrete density/comeagerness results and a notable negative example recast in this framework.
Abstract
We study homeomorphisms and the homeomorphism groups of compact metric spaces using the automorphism groups of projective Fraïssé 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 ${\rm Aut}(\mathbb{P})$.