Dilatation structures with the Radon-Nikodym property

In this paper I explain what is a pair of dilatation structures, one looking down to another. Such a pair of dilatation structures leads to the intrinsic definition of a distribution as a field of topological filters. To any pair of dilatation structures there is an associated notion of differentiability which generalizes the Pansu differentiability. This allows the introduction of the Radon-Nikodym property for dilatation structures, which is the straightforward generalization of the Radon-Nikodym property for Banach spaces. After an introducting section about length metric spaces and metric derivatives, is proved that for a dilatation structure with the Radon-Nikodym property the length of absolutely continuous curves expresses as an integral of the norms of the tangents to the curve, as in Riemannian geometry. Further it is shown that Radon-Nikodym property transfers from any “upper” dilatation structure looking down to a “lower” dilatation structure, theorem \ref{ttransfer}. Im my opinion this result explains intrinsically the fact that absolutely continuous curves in regular sub-Riemannian manifolds are derivable almost everywhere, as proved by Margulis-Mostow, Pansu (for Carnot groups) or Vodopyanov.

💡 Research Summary

The paper introduces a novel geometric framework called a dilatation structure, which generalizes the notion of scaling maps on metric spaces and subsumes classical settings such as Riemannian manifolds, Carnot groups, and sub‑Riemannian manifolds. After a concise review of length spaces and metric derivatives, the author defines a dilatation structure (X, δ) as a family of maps δ⁽ˣ⁾_ε that fix a base point x, satisfy the group‑like law δ⁽ˣ⁾_ε ∘ δ⁽ˣ⁾μ = δ⁽ˣ⁾{εμ}, converge to the identity as ε → 0, and are compatible with the underlying distance.

The central new concept is a pair of dilatation structures (X, δ) and (Y, ε) such that (X, δ) “looks down” on (Y, ε). Intuitively, the scaling of X is finer than that of Y, and the two families can be compared pointwise. This relation allows the author to define a field of topological filters—one filter at each point—that plays the role of a generalized distribution. Unlike classical vector fields, these filters capture the directional information inherent in the hierarchical scaling without requiring linear structure.

With this machinery, a generalized Pansu differentiability is defined. In the classical setting, Pansu differentiability describes the first‑order approximation of a map between Carnot groups by a group homomorphism that respects dilations. Here, the author replaces the group homomorphism by a linear map between the associated filter fields, obtained as a limit of the composition of the two dilation families. This yields a notion of derivative that is intrinsic to the pair (X, δ) ↘ (Y, ε) and does not rely on any ambient linear space.

The paper then extends the Radon‑Nikodym property (RNP) from Banach spaces to dilatation structures. In Banach space theory, a space has the RNP if every σ‑finite measure that is absolutely continuous with respect to Lebesgue measure admits a density function. Analogously, a dilatation structure has the RNP if, for every absolutely continuous curve γ: