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arXiv · 1010.0881

Differentiability of fractal curves

Christoph Bandt·Alexey Kravchenko
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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.

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Christoph Bandt, Alexey Kravchenko. 2010-10-05. Differentiability of fractal curves. https://doi.org/10.1088/0951-7715%2F24%2F10%2F003

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