Dilatation structures I. Fundamentals

A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is apart of a series of papers which show that such a structure allows to do non-commutative analysis, in the sense of differential calculus, on a large class of metric spaces, some of them fractals. We also describe a formal, universal calculus with binary decorated planar trees, which underlies any dilatation structure.

The purpose of this paper is to introduce dilatation structures on metric spaces. A dilatation structure is a concept in between a group and a differential structure. Any metric space (X,d) endowed with a dilatation structure has an associated tangent bundle. The tangent space at a point is a conical group that is the tangent space has a group structure together with a one-parameter group of auto morphisms. Conical groups generalize Carnot groups, i.e nilpotent groups endowed with a graduation. Each dilatation structure leads to a non-commutative differential calculus on the metric space (X, d).