Generating iterated function systems for self-similar sets with a separation condition

2017 ◽  
Vol 237 (2) ◽  
pp. 127-133 ◽  
Author(s):  
Yuanyuan Yao
Keyword(s):  
Iterated Function Systems ◽  
Separation Condition ◽  
Iterated Function ◽  
Self Similar

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

scholarly journals MULTIRESOLUTION CONTROL OF CURVES AND SURFACES WITH A SELF-SIMILAR MODEL

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 271-286 ◽  
Author(s):  
HOUSSAM HNAIDI ◽  
ERIC GUÉRIN ◽  
SAMIR AKKOUCHE
Keyword(s):  
Wavelet Transform ◽  
Control Point ◽  
Iterated Function Systems ◽  
Control Points ◽  
Similar Model ◽  
Surface Theory ◽  
Curves And Surfaces ◽  
Iterated Function ◽  
Resolution Level ◽  
Self Similar

This paper presents two self-similar models that allow the control of curves and surfaces. The first model is based on IFS (Iterated Function Systems) theory and the second on subdivision curve and surface theory. Both of these methods employ the detail concept as in the wavelet transform, and allow the multiresolution control of objects with control points at any resolution level.In the first model, the detail is inserted independently of control points, requiring it to be rotated when applying deformations. In contrast, the second method describes details relative to control points, allowing free control point deformations.Modeling examples of curves and surfaces are presented, showing manipulation facilities of the models.

scholarly journals Dimension approximation of attractors of graph directed IFSs by self-similar sets

2018 ◽  
Vol 167 (01) ◽  
pp. 193-207 ◽  
Author(s):  
ÁBEL FARKAS
Keyword(s):  
Iterated Function System ◽  
Black Box ◽  
Function System ◽  
Separation Condition ◽  
Iterated Function ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Distance Sets

AbstractWe show that for the attractor (K1, . . ., Kq) of a graph directed iterated function system, for each 1 ⩽ j ⩽ q and ϵ > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dimHKj − ϵ < dimHK. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property as a ‘black box’ we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.

Regularity of Hausdorff measure function for conformal dynamical systems

2016 ◽  
Vol 160 (3) ◽  
pp. 537-563 ◽  
Author(s):  
MARIUSZ URBAŃSKI ◽  
ANNA ZDUNIK
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Measure ◽  
Hölder Continuity ◽  
Iterated Function Systems ◽  
Separation Condition ◽  
Holder Continuity ◽  
Measure Function ◽  
Iterated Function ◽  
Strong Separation ◽  
Strong Separation Condition

AbstractWe deal with the question of continuity of numerical values of Hausdorff measures in parametrised families of linear (similarity) and conformal dynamical systems by developing the pioneering work of Lars Olsen and the work [SUZ]. We prove Hölder continuity of the function ascribing to a parameter the numerical value of the Hausdorff measure of either the corresponding limit set or the corresponding Julia set. We consider three cases. Firstly, we consider the case of parametrised families of conformal iterated function systems in $\mathbb{R}$k with k ⩾ 3. Secondly, we consider all linear iterated function systems consisting of similarities in $\mathbb{R}$k with k ⩾ 1. In either of these two cases, the strong separation condition is assumed. In the latter case the Hölder exponent obtained is equal to 1/2. Thirdly, we prove such Hölder continuity for analytic families of conformal expanding repellers in the complex plane $\mathbb{C}$. Furthermore, we prove the Hausdorff measure function to be piecewise real–analytic for families of naturally parametrised linear IFSs in $\mathbb{R}$ satisfying the strong separation condition. On the other hand, we also give an example of a family of linear IFSs in $\mathbb{R}$ for which this function is not even differentiable at some parameters.

scholarly journals Exceptional sets for self-similar fractals in Carnot groups

2010 ◽  
Vol 149 (1) ◽  
pp. 147-172 ◽  
Author(s):  
ZOLTÁN M. BALOGH ◽  
RETO BERGER ◽  
ROBERTO MONTI ◽  
JEREMY T. TYSON
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Iterated Function Systems ◽  
Carnot Groups ◽  
Exceptional Set ◽  
Carathéodory Metric ◽  
Exceptional Sets ◽  
Iterated Function ◽  
Self Similar ◽  
Similarity Dimension

AbstractWe consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

scholarly journals is regular-closed

2018 ◽  
Vol 40 (1) ◽  
pp. 213-220 ◽  
Author(s):  
YUTARO HIMEKI ◽  
YUTAKA ISHII
Keyword(s):  
Rotational Symmetry ◽  
Iterated Function Systems ◽  
Geometric Proof ◽  
Contraction Ratio ◽  
Iterated Function ◽  
Mandelbrot Sets ◽  
Self Similar ◽  
Regular Closed

For each $n\geq 2$, we investigate a family of iterated function systems which is parameterized by a common contraction ratio $s\in \mathbb{D}^{\times }\equiv \{s\in \mathbb{C}:0<|s|<1\}$ and possesses a rotational symmetry of order $n$. Let ${\mathcal{M}}_{n}$ be the locus of contraction ratio $s$ for which the corresponding self-similar set is connected. The purpose of this paper is to show that ${\mathcal{M}}_{n}$ is regular-closed, that is, $\overline{\text{int}\,{\mathcal{M}}_{n}}={\mathcal{M}}_{n}$ holds for $n\geq 4$. This gives a new result for $n=4$ and a simple geometric proof of the previously known result by Bandt and Hung [Fractal $n$-gons and their Mandelbrot sets. Nonlinearity 21 (2008), 2653–2670] for $n\geq 5$.

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