Topology of Attractors
by
Marcy Barge
Montana State University
Coauthors: Beverly Diamond (College of Charleston)

The transitive, nonseparating, planar attractors whose existence has recently been established by Benedicks and Carleson all occur when the unstable manifold of a fixed hyperbolic saddle (the closure of which equals the attractor) is nearly tangent with the stable manifold of the saddle. Globally, these attractors are indecomposable continua. Unlike one-dimensional hyperbolic attractors, which are everywhere locally homeomorphic with the product of a Cantor set and an arc, these nonseparating attractors may have a very rich local structure as well.

Theorem: Suppose that F: R² --> R² is a C^\infty diffeomorphism with fixed hyperbolic saddle p. Suppose also that a branch, W₊^u(p), of the unstable manifold of p has a (same-sided) nondegenerate tangency with the stable manifold of p and suppose that the logs of the eigenvalues of the Jacobian of F at p are independent over Q. Then in every neighborhood of every point of the closure of W₊^u(p) there is, in clW₊^u(p), a homeomorphic copy of every continuum that can be expressed as an inverse limit of a sequence of unimodal bonding maps.

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-33.