How strange can an attractor for a dynamical system in a 3-manifold look?

✍ Scribed by J.J. Sánchez-Gabites

• Book ID: 103849112
• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 469 KB
• Volume: 74
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2011.05.095

✦ Synopsis

The aim of this paper is to characterise those compact subsets K of 3-manifolds M that are (stable and not necessarily global) attractors for some flow on M. We will show that it is the topology of M -K , rather than that of K , the one that plays a relevant role in this problem.

A necessary and sufficient condition for a set K to be an attractor is that it must be an ''almost tame'' subset of M in a sense made precise under the equivalent notions of ''weakly tame'' and ''tamely embedded up to shape'', defined in the paper. These are complemented by a further equivalent condition, ''algebraic tameness'', which has the advantage of being checkable by explicit computation.

A final section of the paper is devoted to a partial analysis of the same question when one replaces flows by discrete dynamical systems.