Homeomorphisms of continua through projective Fraïssé limits

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 $\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(P)$ of all homeomorphisms of the pseudoarc $P$ using the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(\mathbb{P})$ of the pre-pseudoarc $\mathbb{P}$. Strengthening results from the literature, we show that the diagonal conjugacy action of $\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(P)$ on $\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(P{)}^{\mathbb{N}}$ has a dense orbit. In our second application, we show that there exists a homeomorphism of $P$ that is not conjugate in $\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(P)$ to an element of $\mathrm{A}\mathrm{u}\mathrm{t}(\mathbb{P})$.