Contraction of a Generalized Metric Structure

Guy Wallet (1)

1 MIA - Mathématiques, Image et Applications - EA 3165

In some scientific fields, a scaling is able to modify the topology of an observed object. Our goal in the present work is to introduce a new formalism adapted to the mathematical representation of this kind of phenomenon. To this end, we introduce a new metric structure - the galactic spaces - which depends on an ordered field extension of R. Moreover, some natural transformations of the category of galactic spaces, the contractions, are of particular interest: they generalize usual homotheties, they have a ratio which may be an infinitesimal, they are able to modify the topology and they satisfy a nice composition rule. With the help of nonstandard extensions we can associate to any metric space an infinite family of galactic spaces; lastly, we study some limit properties of this family.

Mots clés: Scaling; contraction; generalized distance; galactic space

Domaines: Topologie géométrique [math.GT]; Géométrie métrique [math.MG]

Guy Wallet. Contraction of a Generalized Metric Structure. 2008. ⟨hal-00345329⟩