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Aseev, V. V.; Tetenov, Andrei; Kravchenko, A. S.

doi:10.1023/a:1023848327898

Cited by 27 publications

(17 citation statements)

References 5 publications

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“…As it was proved by Aseev et al [1], there is unique continuous f : [0, 1] → γ such that f (t i ) = z i (i = 1, ..., m), and for any i = 1, ..., m and t…”

mentioning

confidence: 79%

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On the morphisms of fractal curves that increase their smoothness

Kamalutdinov,

Sorokina

2015

Preprint

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We propose a construction which transforms a self-similar zipper in R n to a self-affine zipper R n+1 whose attractor is a smooth curve. MSC classification: Primary 28A80Smooth self-affine curves may be constructed in two ways: 1. As it was proved by M. Barnsley [3], for some special types of fractal interpolation functions their integration gives differentiable functions whose graphs are self-affine; 2. The other way is to use the algorythm of building self-affine curves proposed by Alexey Kravchenko in 2005. [5] We propose one more construction, which transforms a self-similar zipper to a self-affine zipper whose attractor is a smooth curve.

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“…As it was proved by Aseev et al [1], there is unique continuous f : [0, 1] → γ such that f (t i ) = z i (i = 1, ..., m), and for any i = 1, ..., m and t…”

mentioning

confidence: 79%

“…An attractor of any system exists and unique due to Hutchinson theorem [4]. Attractor of any zipper is arcwise connected and locally arcwise connected [1].…”

mentioning

confidence: 99%

On the morphisms of fractal curves that increase their smoothness

Kamalutdinov,

Sorokina

2015

Preprint

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“…The proof is similar to (cf. [1,Lemma 1]). First, we define the functionf which is a surjection of the dense subset G S (V P )⊂K to the dense subset G S (V P )⊂K .…”

Section: δ-Deformations Of Contractible Polygonal Systemsmentioning

confidence: 99%

“…Then S b1 (γ 1 )⊂γ 1 and S b2 (γ 2 )⊂γ 2 . From [1,Lemma 3.1] it follows that if γ 1 ∩ γ 2 = {C}, then λ 1 = λ 2 .…”

Section: Parameter Matching Theoremmentioning

confidence: 99%

On deformation of polygonal dendrites preserving the intersection graph

Drozdov

Samuel²,

Tetenov

2020

Art Discrete Appl. Math.

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Let S = S 1 , ..., S m be a system of contracting similarities of R 2 . The attractor K(S) of the system S is a non-empty compact set satisfying K = S 1 (K) ∪ ... ∪ S m (K). We consider contractible polygonal systems S which are defined by a finite family of polygons whose intersection graph is a tree and therefore the attractor K(S) is a dendrite. We find conditions under which a deformation S of a contractible polygonal system S has the same intersection graph and therefore the attractor K(S ) is a self-similar dendrite which is isomorphic to the attractor K of the system S.

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“…A method of construction of self-similar curves, used by many authors [15,13,9] was studied in 2002 by V.V.Aseev [1] as a zipper construction. This construction proved to be an efficient tool in the investigation of geometrical properties of self-similar curves and continua [2]. Its graph-directed version was introduced by the author in 2006 and was called a multizipper construction; it gives a complete description of self-similar Jordan arcs in R d [16, Theorem 4.1]:…”

Section: Graph-directed Systems Of Contraction Similaritiesmentioning

confidence: 99%