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Movability and limits of polyhedra

V. Laguna;  M. Moron;  Nhu Nguyen;  J. Sanjurjo — 1993

Fundamenta Mathematicae

We define a metric d S , called the shape metric, on the hyperspace 2 X of all non-empty compact subsets of a metric space X. Using it we prove that a compactum X in the Hilbert cube is movable if and only if X is the limit of a sequence of polyhedra in the shape metric. This fact is applied to show that the hyperspace ( 2 ℝ 2 , dS) i s s e p a r a b l e . O n t h e o t h e r h a n d , w e g i v e a n e x a m p l e s h o w i n g t h a t 2ℝ2 i s n o t s e p a r a b l e i n t h e f u n d a m e n t a l m e t r i c i n t r o d u c e d b y B o r s u k .

Topology and dynamics of unstable attractors

M. A. Morón;  J. J. Sánchez-Gabites;  J. M. R. Sanjurjo — 2007

Fundamenta Mathematicae

This article aims to explore the theory of unstable attractors with topological tools. A short topological analysis of the isolating blocks for unstable attractors with no external explosions leads quickly to sharp results about their shapes and some hints on how Conley's index is related to stability. Then the setting is specialized to the case of flows in ℝⁿ, where unstable attractors are seen to be dynamically complex since they must have external explosions.

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