NON-DIFFERENTIABLE DEFORMATIONS OF ℝn

Abstract

Many problems of physics or biology involve very irregular objects like the rugged surface of a malignant cell nucleus or the structure of space-time at the atomic scale. We define and study non-differentiable deformations of the classical Cartesian space ℝn which can be viewed as the basic bricks to construct irregular objects. They are obtained by taking the topological product of n-graphs of nowhere differentiable real valued functions. Our point of view is to replace the study of a non-differentiable function by the dynamical study of a one-parameter family of smooth regularization of this function. In particular, this allows us to construct a one-parameter family of smooth coordinates systems on non-differentiable deformations of ℝn, which depend on the smoothing parameter via an explicit differential equation called a scale law. Deformations of ℝn are examples of a new class of geometrical objects called scale manifolds which are defined in this paper. As an application, we derive rigorously the main results of the scale-relativity theory developed by Nottale in the framework of a scale space-time manifold.