Extensions of Homeomorphisms of Continua
by
James L. Kassebaum
Montana State University

Barge and Martin have shown that the inverse limit of any interval map may be realized as an attractor of a planar homeomorphism H in such a way that H restricted to the attractor is conjugate to the induced homeomorphism on the inverse limit. Since the inverse limit of any interval map is chainable, a logical question is: "given a chainable continuum X and a homeomorphism g: X --> X, is there an embedding \Psi: X --> R² so that there exists a planar homeomorphism G such that \Psi(X) is an attractor for G and G|_\Psi(X) is conjugate to g?". An example showing that the conjecture is false is given. Moreover, it is shown that given any nondegenerate planar continuum \Lambda and homeomorphism f: \Lambda --> \Lambda, which is not the identity, there is an embedding \phi: \Lambda --> R³ such that \phi o f o \phi^-1 cannot be extended to all of R³.

Date received: January 31, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaa-30.