On Selfsimilar Jordan Curves on the Plane

  title={On Selfsimilar Jordan Curves on the Plane},
  author={V. Aseev and A. Tetenov and A. Kravchenko},
  journal={Siberian Mathematical Journal},
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 the signature, the mapping Sj takes the pair {x0,xn} either into the pair {xj-1,xj} (if sj =0) or into the pair {xj, xj-1} (if sj=1). We describe the situations in which the Jordan property of such attractor implies that the attractor has bounded turning, i.e., is a quasiconformal image of an… Expand
On the Self-Similar Jordan Arcs Admitting Structure Parametrization
AbstractWe study the attractors γ of a finite system $$S$$ of contraction similarities Sj (j = 1,..., m) in ℝd which are Jordan arcs. We prove that if a system $$S$$ possesses a structureExpand
The fractal “Frog”
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On transverse hyperplanes to self-similar Jordan arcs.
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-Similar Jordan Arcs Which Do Not Satisfy OSC
It was proved in 2007 by C.Bandt and H.Rao that if a system $S = \{S_1 , ..., S_m \}$ of contraction similarities in $R^2$ with a connected attractor $K$ has the finite intersection property, then itExpand
Iterated function system quasiarcs
We consider a class of iterated function systems (IFSs) of contracting similarities of $R^n$, introduced by Hutchinson, for which the invariant set possesses a natural H\"older continuousExpand
Self-similar Jordan arcs and the graph directed systems of similarities
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On dendrites, generated by polyhedral systems and their ramification points
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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 andExpand
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We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases manyExpand


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Given a self-similar set K in Rs we prove that the strong open set condition and the open set condition are both equivalent to Ha (K) > 0, where a is the similarity dimension of K and Ha denotes theExpand
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For self-similar sets with nonoverlapping pieces, Hausdorff dimension and measure are easily determined. We express absence of overlap in terms of discontinuous action of a family of similitudes,Expand
Introduction to Complex Analysis
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