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Differentiability of fractal curves

A self-similar set that spans [dbl-struck R] super(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 cur...

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Bibliographic Details
Published in:Nonlinearity 2011-10, Vol.24 (10), p.2717-2728
Main Authors: BANDT, Christoph, KRAVCHENKO, Aleksey
Format: Article
Language:English
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Summary:A self-similar set that spans [dbl-struck R] super(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.
ISSN:0951-7715
1361-6544
DOI:10.1088/0951-7715/24/10/003