Differentiability of fractal curves
Abstract
A self-similar set that spans { R}^n can have no tangent hyperplane at any single point. There are lots of smooth self-affine curves, however. 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 line segments and parabolic arcs.
- Publication:
-
Nonlinearity
- Pub Date:
- October 2011
- DOI:
- 10.1088/0951-7715/24/10/003
- arXiv:
- arXiv:1010.0881
- Bibcode:
- 2011Nonli..24.2717B
- Keywords:
-
- Mathematics - Dynamical Systems;
- Mathematics - Metric Geometry;
- 28A80;
- 26A27
- E-Print:
- doi:10.1088/0951-7715/24/10/003