Glasnik Matematicki, Vol. 42, No.1 (2007), 117-130.

SHAPE PROPERTIES OF THE BOUNDARY OF ATTRACTORS

J. J. Sánchez-Gabites and José M. R. Sanjurjo

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.

2000 Mathematics Subject Classification.   54H20, 55P55, 58F12.

Key words and phrases.   Dynamical systems, attractors, boundary of attractors, shape.

DOI: 10.3336/gm.42.1.10

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