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 {Pₙ,gₙ} where Pₙ = P is a fixed ANR and gₙ = g: P → P is a fixed bonding map.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-1-4,
author = {Francisco R. Ruiz del Portal and Jos\'e M. Salazar},
title = {Shape index in metric spaces},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {47-62},
zbl = {1082.37019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-1-4}
}
Francisco R. Ruiz del Portal; José M. Salazar. Shape index in metric spaces. Fundamenta Mathematicae, Tome 177 (2003) pp. 47-62. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-1-4/