Original scientific paper

https://doi.org/10.3336/gm.42.1.10

Shape properties of the boundary of attractors

J. J. Sánchez-Gabites ; Departamento de Geometria y Topologia, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
José M. R. Sanjurjo ; Departamento de Geometria y Topologia, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Abstract

Let M be a locally compact metric space endowed with a continuous flow φ : M × R → M. Assume that K is a stable attractor for φ and P A(K) is a compact positively invariant neighbourhood of K contained in its basin of attraction. Then it is known that the inclusion K P is a shape equivalence and the question we address here is whether there exists some relation between the shapes of ∂K and ∂P. The general answer is negative, as shown by example, but under certain hypotheses on K the shape domination Sh(∂K) ≥ Sh(∂P) or even the equality Sh(∂K) = Sh(∂P) hold. However we also put under study interesting situations where those hypotheses are not satisfied, albeit other techniques such as Lefschetz's duality render results relevant to our question.

Keywords

Dynamical systems; attractors; boundary of attractors; shape

Hrčak ID:

12889

URI

https://hrcak.srce.hr/12889

Publication date:

12.6.2007.

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