Most Maps of the Pseudo-Arc are Homeomorphisms

Abstract

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 H(P) of maps of the pseudo-arc into itself which are homeomorphisms onto their images is a dense Gs in M(P). (2) Every homeomorphism of the pseudo-arc onto itself is a product of e-homeomorphisms.(3) There exists a nonidentity homeomorphism of the pseudo-arc with an infinite sequence of pth roots. (4) Every map between chainable continua can be lifted to a homeomorphism of pseudo-arcs.We will investigate certain properties of maps and homeomorphisms of the pseudo-arc and other chainable continua. The pseudo-arc is characterized [2] as a nondegenerate, hereditarily indecomposable, chainable continuum. In most cases we will be defining maps or investigating their behavior in terms of functions between defining sequences of chains and patterns. For definitions of chain, link, pattern, amalgamation, and crooked, and for basic properties of these, the reader is referred to [1, 2, 9, and 12]. Unless otherwise specified all maps are surjections.Some specific facts we will make use of are the following.Lemma 1 [2]. If X is a nondegenerate chainable continuum such that for each chain C covering X and e > 0 there is a chain D of mesh less than e which covers X and is crooked in C, then X is a pseudo-arc.

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