Most maps of the pseudo-arc are homeomorphisms

Abstract:

We prove the following results. (1) If $M(P)$ is the space of maps of the pseudo-arc into itself with the sup metric, then the subset $\hat H(P)$ of maps of the pseudo-arc into itself which are homeomorphisms onto their images is a dense ${G_\delta }$ in $M(P)$. (2) Every homeomorphism of the pseudo-arc onto itself is a product of $\in$-homeomorphisms. (3) There exists a nonidentity homeomorphism of the pseudo-arc with an infinite sequence of $p$th roots. (4) Every map between chainable continua can be lifted to a homeomorphism of pseudo-arcs.