Differentiability of fractal curves
Dynamical Systems
2012-04-18 v1 Metric Geometry
Abstract
While self-similar sets have no tangents at any single point, self-affine curves can be smooth. We consider plane self-affine curves without double points and with two pieces. There is an open subset of parameter space for which the curve is differentiable at all points except for a countable set. For a parameter set of codimension one, the curve is continuously differentiable. However, there are no twice differentiable self-affine curves in the plane, except for parabolic arcs.
Keywords
Cite
@article{arxiv.1010.0881,
title = {Differentiability of fractal curves},
author = {Christoph Bandt and Alexey Kravchenko},
journal= {arXiv preprint arXiv:1010.0881},
year = {2012}
}