# Differentiability of fractal curves

@article{Bandt2011DifferentiabilityOF, title={Differentiability of fractal curves}, author={Christoph Bandt and A. S. Kravchenko}, journal={Nonlinearity}, year={2011}, volume={24}, pages={2717-2728} }

A self-similar set that spans 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… Expand

#### 6 Citations

Pointwise regularity of parameterized affine zipper fractal curves

- Mathematics
- 2016

We study the pointwise regularity of zipper fractal curves generated by affine mappings. Under the assumption of dominated splitting of index-1, we calculate the Hausdorff dimension of the level sets… Expand

On transverse hyperplanes to self-similar Jordan arcs.

- Mathematics
- 2014

We consider self-similar Jordan arcs γ in \(\mathbb{R}^{d}\), different from a line segment and show that they cannot be projected to a line bijectively. Moreover, we show that the set of points x ∈… Expand

Self-affine sets in analytic curves and algebraic surfaces

- Mathematics
- 2016

We characterize analytic curves that contain non-trivial self-affine sets. We also prove that compact algebraic surfaces cannot contain non-trivial self-affine sets.

Novel method of fractal approximation

- Mathematics
- 2012

We introduce new method of optimization for finding free parameters of affine iterated function systems (IFS), which are used for fractal approximation. We provide the comparison of effectiveness of… Expand

On Weak Separation Property for Affine Fractal Functions

- Mathematics
- 2015

We show that a fractal affine function $f(x)$ defined by a system $\mathcal S$ which does not satisfy weak separation property is a quadratic function.

ON WEAK SEPARATION PROPERTY FOR AFFINE FRACTAL FUNCTIONS

- 2015

We show that a fractal affine function f(x) defined by a system S which does not satisfy weak separation property is a quadratic function.

#### References

SHOWING 1-10 OF 12 REFERENCES

Topology and separation of self-similar fractals in the plane

- Mathematics
- 2007

Even though the open set condition (OSC) is generally accepted as the right condition to control overlaps of self-similar sets, it seems unclear how it relates to the actual size of the overlap. For… Expand

On the geometric structure of the limit set of conformal iterated function systems

- Mathematics
- 2003

We consider infinite conformal iterated function systems on R. We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some l–dimensional C–submanifold… Expand

Dimension and measures on sub-self-affine sets

- Mathematics
- 2010

We show that in a typical sub-self-affine set, the Hausdorff and the Minkowski dimensions coincide and equal the zero of an appropriate topological pressure. This gives a partial positive answer to… Expand

On Selfsimilar Jordan Curves on the Plane

- Mathematics
- 2003

We study the attractors of a finite system of planar contraction similarities Sj (j=1,...,n) satisfying the coupling condition: for a set {x0,...,xn} of points and a binary vector (s1,...,sn), called… Expand

Additive functions of intervals and Hausdorff measure

- Mathematics
- 1946

Consider bounded sets of points in a Euclidean space R q of q dimensions. Let h(t) be a continuous increasing function, positive for t >0, and such that h (0) = 0. Then the Hausdroff measure h–mE of… Expand

Vilppolainen, Dimension and measures on sub-self-affine sets, Monatshefte Math

- 2010

Smooth self-affine curves (in Russian)

- Preprint No. 161, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
- 2005

A

- V. Tetenov and A. S. Kravchenko,On Selfsimilar Jordan Curves on the Plane, Siberian Math. J. 44, No. 3
- 2003