Most maps of the pseudo-arc are homeomorphisms

Abstract

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