Dilatation structures with the Radon-Nikodym property

The notion of a dilatation structure stemmed out from my efforts to understand basic results in sub-Riemannian geometry, especially the last section of the paper by Belläıche [2] and the intrinsic point of view of Gromov [5]. In these papers, as in other articles devoted to sub-Riemannian geometry, fundamental results admiting an intrinsic formulation were proved using differential geometry tools, which are in my opinion not intrinsic to sub-Riemannian geometry. Therefore I tried to find a self-contained frame in which sub-Riemannian geometry would be a model, if we use the same manner of speaking as in the case of hyperbolic geometry (with its self-contained collection of axioms) and the Poincaré disk as a model of hyperbolic geometry. An outcome of this effort are the notions of a dilatation structure and a pair of dilatation structures, one looking down to another. To the first notion are dedicated the papers [3], [4] (the second paper treating about a ”linear” version of a generalized dilatation structure, corresponding to Carnot groups or more general contractible groups). As it seems now, dilatation structures are a valuable notion by itself, with possible field of application strictly containing sub-Riemannian geometry, but also ultrametric spaces or contractible groups. A dilatation structure encodes the approximate self-similarity of a metric space and it induces non associative but approximately associative operations on the metric space, as well as a tangent bundle (in the metric sense) with group operations in each fiber (tangent space to a point). In this paper I explain what is a pair of dilatation structures, one looking down to another, see definition 3.5. Such a pair of dilatation structures leads to the intrinsic definition of a distribution as a field of topological filters, definition 3.6. To any pair of dilatation structures there is an associated notion of differentiability which generalizes the Pansu differentiability [8]. This allows the introduction of