Geometric differentiability of Riemann's non-differentiable function

Abstract

Riemann's non-differentiable function is a classic example of a continuous function which is almost nowhere differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a geometric interpretation, so questions on its geometric regularity arise. This point of view is developed in the context of the evolution of vortex filaments, modelled by the Vortex Filament Equation or the binormal flow, in which a generalisation of Riemann's function to the complex plane can be regarded as the trajectory of a particle. The objective of this document is to show that the trajectory represented by its image does not have a tangent anywhere. For that, we discuss several concepts of tangent vectors in view of the set's irregularity.

27 pages, 6 figures. v2: Typos corrected, references added. v3: Accepted manuscript

References in corpus (2)

• Vortex Filament Equation for a Regular Polygon
• Some geometric properties of Riemann's non-differentiable function

Cited by in corpus (6)

• Riemann's non-differentiable function and the binormal curvature flow
• Convergence over fractals for the periodic Schrödinger equation
• Some geometric properties of Riemann's non-differentiable function
• The pointwise behavior of Riemann's function
• Intermittency of Riemann's non-differentiable function through the fourth-order flatness
• An analytical study of flatness and intermittency through Riemann's non-differentiable functions