Shape index in metric spaces

Francisco R. Ruiz del Portal¹ ; José M. Salazar²

¹ Departamento de Geometría y Topología Facultad de Matemáticas Universidad Complutense de Madrid Madrid 28040, Spain
² Departamento de Matemáticas Universidad de Alcalá Alcalá de Henares Madrid 28871, Spain

Fundamenta Mathematicae, Tome 176 (2003) no. 1, pp. 47-62

Résumé

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 $g_n=g: P \rightarrow P$ is a fixed bonding map.

Détail

DOI : 10.4064/fm176-1-4

Keywords: extend shape index introduced robbin salamon mrozek locally defined maps metric spaces index additive construction answers affirmative questions posed mrozek prove shape index cannot arbitrarily complicated shapes q adic solenoids appear shape indices natural modifications smales horseshoes there compact isolated invariant set locally defined map locally compact metric anr whose shape index shape generalized solenoid maps defined locally compact metric anrs shape index always computed hilbert cube consequently shape index shape inverse limit sequence where fixed anr rightarrow fixed bonding map