2010
DOI: 10.1142/s0218348x1000510x
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Remarks on Self-Affine Fractals With Polytope Convex Hulls

Abstract: Suppose that the set T = {T 1 , T 2 , . . . , T q } of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d 1 , . . . , d q } ⊂ R n , there exists a unique non-empty compact, which is typically a fractal set. We use the infinite digit expansions of the points of F to give simple necessary and sufficient conditions for the convex hull of F to be a polytope. Additionally, we present a technique to determine the vertices of such polytopes. These answer some of the related quest… Show more

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Cited by 8 publications
(10 citation statements)
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“…The work most relevant to this paper was done by Kırat and Koçyigit [11,12,13] focusing essentially on Problem 1.1 for self-affine sets. They devise a constructive terminating algorithm for finding the convex hull of IFS fractals when the factors are equal, and a non-terminating method when they are not.…”
Section: The Problem and Former Resultsmentioning
confidence: 99%
See 2 more Smart Citations
“…The work most relevant to this paper was done by Kırat and Koçyigit [11,12,13] focusing essentially on Problem 1.1 for self-affine sets. They devise a constructive terminating algorithm for finding the convex hull of IFS fractals when the factors are equal, and a non-terminating method when they are not.…”
Section: The Problem and Former Resultsmentioning
confidence: 99%
“…The above implies that having equal IFS rotation angles 2πN/M , it is sufficient to generate all M -th level periodic points (computed using Corollary 1.2), as their convex hull will be that of the fractal. This is a simplified version of the method presented by Kırat and Koçyigit [11] for the special case of planar equiangular IFS fractals.…”
Section: A Methods For Equiangular Fractals Of Unitymentioning
confidence: 99%
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“…( In [26] was proved the following result about the extreme points of conv(F m ) for any m > n, where n This implies conv(T (L, F 1 )) is equal to (L m − id) −1 conv(F m ) for all m > n.…”
Section: 2mentioning
confidence: 99%
“…[26, Theorem 4.8] If | Ext(conv(F n ))| = | Ext(conv(F n+1 ))|,then all the extreme points of conv(T (L, F 1 )) are of the form j>0 L −(n+1)j n i=0 L i (f i ) , with n i=0 L i (f i ) being an extreme point of conv(F n+1 ).…”
mentioning
confidence: 99%