The Structure of Atoms, Part II
by
Yaki Sternfeld
University of Haifa
Coauthors: Richard Ball (University of Denver), James Hagler (University of Haifa)

Let X be an atom (= hereditarily indecomposable continuum). A continuous map W:2^X --> [0, 1] is a Whitney map if W( { x}) = 0 for every singleton { x} , W( X) = 1 and A\subsetneqq B implies W( A) < W( B) . Define a metric \rho on X by letting \rho( x, y) = W(A_xy) where A_{x, y} is the (unique) minimal subcontinuum of X which contain x and y. \rho is an ultrametric and the topology of ( X, \rho) is stronger than the original topology of X. The \rho closed balls C( x, r) = { y in X:\rho(x, y) <= r} coincide with the subcontinua of X. (C(x, r) is the unique subcontinuum of X which contains x and has Whitney value r.) It is proved that for every two (nontrivial) atoms and any Whitney maps on them, the corresponding ultrametric spaces are isometric. This implies in particular that the combinatorial structure of subcontinua is identical in all atoms.

The set M( X) of all monotone upper semi-continuous decompositions of X is a lattice when ordered by refinement. It is proved that for two atoms X and Y, M( X) is lattice isomorphic to M( Y) if and only if X is homeomorphic to Y.

The results of this study will be presented in two talks, the first by J. Hagler and the second by Y. Sternfeld.

Date received: June 20, 1997

Copyright © 1997 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 # caao-13.