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Shape index in metric spaces

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Romero Ruiz del Portal, Francisco and Salazar, J. M. (2003) Shape index in metric spaces. Fundamenta Mathematicae, 176 (1). pp. 47-62. ISSN 0016-2736

Official URL: http://journals.impan.gov.pl/fm/



Abstract

We extend the shape index, introduced by Robbin and Salamon and Mrozek, to locally defined maps in metric spaces. We show that this index is additive. Thus our construction answers in the affirmative two questions posed by Mrozek in [12]. We also prove that the shape index cannot be arbitrarily complicated: the shapes of q -adic solenoids appear as shape indices in natural modifications of Smale's horseshoes but there is not any compact isolated invariant set for any locally defined map in a locally compact metric ANR whose shape index is the shape of a generalized solenoid. We also show that, for maps defined in locally compact metric ANRs, the shape index can always be computed in the Hilbert cube. Consequently, the shape index is the shape of the inverse limit of a sequence {P n ,g n } where P n =P is a fixed ANR and g n =g:P→P is a fixed bonding map.


Item Type:Article
Uncontrolled Keywords:Semidynamical systems; Conley index; Shape theory
Subjects:Sciences > Mathematics > Topology
ID Code:19892
Deposited On:11 Feb 2013 15:04
Last Modified:12 Dec 2018 15:13

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