Every Attractor of a Flow on a Manifold has the Shape of a Finite Polyhedron

Unstable manifold, Conley index and fixed points of flows

“…For a complete treatment of shape theory we refer the reader to [7,12,13,27,26,38] . The use of shape in dynamics is illustrated by the papers [18,15,19,21,24,32,33,36]. For information about basic aspects of dynamical systems we recommend [5,34,44] and for algebraic topology the books written by Hatcher [22] and Spanier [42] are very useful.…”

A compactum that cannot be an attractor of a self-map on a manifold

“…We may generalize still further and consider arbitrary self-maps / : X -> X instead of homeomorphisms. A compact subset AC X with f(A) = A is an attractor of / if there is a neighborhood U of A in X, such that f(U) C U and for each neighborhood V of A in X there is n e N with fn(U) C V; this implies fm(U) C V for all m>n. In this situation the classification of attractors seems to be much more complicated than in [2], but at least we are able to produce an example of a one-dimensional compactum that can never be an attractor of a self-map.It is well known that the dyadic solenoid is an attractor of a homeomorphism of a three-dimensional manifold (see [4; 2, Example 3]). The dyadic solenoid is the limit of an inverse sequence of circles Sxn^ = {z e C | |z| = 1}, and the bonding map gn : 5'(1n+1) -► Sxn^ is given by g"(z) = z2.…”

Topology and dynamics of unstable attractors

“…In the realm of continuous dynamical systems the notion of attractor plays a very important role because it captures the long term evolution of the system in question, and therefore it seems important to study the structure, both dynamical and topological, of these objects. Very sharp results, mainly concerning the shape, have been obtained for stable attractors (see for example [6], [16], [18], [26], [27]), but when it comes to unstable attractors not much is known, and in fact the bibliography concerning the subject is quite scarce (essentially [1]- [4], [22], [29]). Let us remark that papers [1] and [29] use Milnor's notion of attractor which is slightly different from ours.…”