C*-Algebras Associated with Mauldin–Williams Graphs

An orbital picture is a mathematical structure depicting the path of an object under Iterated Function System. Orbital and V-variable orbital pictures initially developed by Barnsley (2006) have utmost importance in computer graphics, image compression, biological modeling and other areas of fractal geometry. These pictures have been generated for linear and contractive transformations using function and superior iterative procedures. In this paper, the authors introduce the role of superior iterative procedure to find the orbital picture under an IFS consisting of non-contractive or non-expansive transformations. A mild comparison of the computed figures indicates the usefulness of study in computational mathematics and fractal image processing. A modified algorithm along with program code is given to compute a 2-variable superior orbital picture.

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Lattice self-similar sets on the real line are not Minkowski measurable

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.26 ◽

2018 ◽

Vol 40 (1) ◽

pp. 221-232

Author(s):

SABRINA KOMBRINK ◽

STEFFEN WINTER

Keyword(s):

Real Line ◽

Iterated Function System ◽

Function System ◽

Open Set Condition ◽

The Real ◽

Open Set ◽

Puzzle Piece ◽

Iterated Function ◽

Self Similar ◽

Special Classes

We show that any non-trivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this has only been known for special classes of such sets. Thus, we provide the last puzzle-piece in proving that under the OSC a non-trivial self-similar subset of the real line is Minkowski measurable if and only if it is invariant under a non-lattice IFS, a 25-year-old conjecture.

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Fractals in the Large

Canadian Journal of Mathematics ◽

10.4153/cjm-1998-036-5 ◽

1998 ◽

Vol 50 (3) ◽

pp. 638-657 ◽

Cited By ~ 28

Author(s):

Robert S. Strichartz

Keyword(s):

Invariant Measure ◽

Power Series ◽

Metric Space ◽

Iterated Function System ◽

Invariant Set ◽

Invariant Sets ◽

Iterated Function ◽

Positive Weights ◽

Self Similar ◽

Expansive Maps

AbstractA reverse iterated function system (r.i.f.s.) is defined to be a set of expansive maps ﹛T1,…, Tm﹜ on a discrete metric space M. An invariant set F is defined to be a set satisfying , and an invariant measure μ is defined to be a solution of for positive weights pj. The structure and basic properties of such invariant sets and measures is described, and some examples are given. A blowup ℱ of a self-similar set F in ℝn is defined to be the union of an increasing sequence of sets, each similar to F. We give a general construction of blowups, and show that under certain hypotheses a blowup is the sum set of F with an invariant set for a r.i.f.s. Some examples of blowups of familiar fractals are described. If μ is an invariant measure on ℤ+ for a linear r.i.f.s., we describe the behavior of its analytic transform, the power series on the unit disc.

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REVERSE ITERATED FUNCTION SYSTEM AND DIMENSION OF DISCRETE FRACTALS

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270800097x ◽

2009 ◽

Vol 79 (1) ◽

pp. 37-47 ◽

Cited By ~ 1

Author(s):

QI-RONG DENG

Keyword(s):

Iterated Function System ◽

Invariant Set ◽

Function System ◽

Computation Method ◽

Iterated Function ◽

Expansive Maps

AbstractA reverse iterated function system is defined as a family of expansive maps {T1,T2,…,Tm} on a uniformly discrete set $M\subset \Bbb {R}^d$. An invariant set is defined to be a nonempty set $F\subseteq M$ satisfying F=⋃ j=1mTj(F). A computation method for the dimension of the invariant set is given and some questions asked by Strichartz are answered.

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Orbit of an Image Under Iterated System II

Investigations into Living Systems, Artificial Life, and Real-World Solutions ◽

10.4018/978-1-4666-3890-7.ch014 ◽

2013 ◽

pp. 154-168

Author(s):

S. L. Singh ◽

S. N. Mishra ◽

Sarika Jain

Keyword(s):

Fractal Geometry ◽

Iterative Procedure ◽

Iterated Function System ◽

Mathematical Structure ◽

Computational Mathematics ◽

Fractal Image ◽

Iterated Function ◽

Iterative Procedures ◽

Modified Algorithm

An orbital picture is a mathematical structure depicting the path of an object under Iterated Function System. Orbital and V-variable orbital pictures initially developed by Barnsley (2006) have utmost importance in computer graphics, image compression, biological modeling and other areas of fractal geometry. These pictures have been generated for linear and contractive transformations using function and superior iterative procedures. In this paper, the authors introduce the role of superior iterative procedure to find the orbital picture under an IFS consisting of non-contractive or non-expansive transformations. A mild comparison of the computed figures indicates the usefulness of study in computational mathematics and fractal image processing. A modified algorithm along with program code is given to compute a 2-variable superior orbital picture.

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The Chaos Game on a General Iterated Function System from a Topological Point of View

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414501399 ◽

2014 ◽

Vol 24 (11) ◽

pp. 1450139 ◽

Cited By ~ 7

Author(s):

Michael F. Barnsley ◽

Krzysztof Leśniak

Keyword(s):

Stochastic Process ◽

Iterated Function System ◽

Almost Sure Convergence ◽

Baire Category ◽

Point Of View ◽

Function System ◽

Iterated Function ◽

Chaos Game

We investigate combinatorial issues relating to the use of random orbit approximations to the attractor of an iterated function system with the aim of clarifying the role of the stochastic process during the generation of the orbit. A Baire category counterpart of almost sure convergence is presented.

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Inhomogeneous self-similar sets with overlaps

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.13 ◽

2017 ◽

Vol 39 (1) ◽

pp. 1-18 ◽

Cited By ~ 2

Author(s):

SIMON BAKER ◽

JONATHAN M. FRASER ◽

ANDRÁS MÁTHÉ

Keyword(s):

Final Section ◽

Iterated Function System ◽

Box Dimension ◽

Open Set Condition ◽

Open Set ◽

Bernoulli Convolutions ◽

Iterated Function ◽

Different Types ◽

Self Similar ◽

Weak Separation

It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this ‘expected formula’ does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of a spectral gap property that holds for certain subgroups of $\text{SO}(d)$ for $d\geq 3$. We also obtain new upper bounds, derived using sumsets, for the upper box dimension of an inhomogeneous self-similar set which hold in general. Moreover, our counterexamples demonstrate that these bounds are optimal. In the final section we show that if the weak separation property is satisfied, that is, the overlaps are controllable, then the ‘expected formula’ does hold.

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A strongly irreducible affine iterated function system with two invariant measures of maximal dimension

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.107 ◽

2020 ◽

pp. 1-22

Author(s):

IAN D. MORRIS ◽

CAGRI SERT

Keyword(s):

Invariant Measures ◽

Iterated Function System ◽

Iterated Function Systems ◽

Finite Union ◽

Function System ◽

Maximal Dimension ◽

Similar Measure ◽

Open Set ◽

Iterated Function ◽

Self Similar

Abstract A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.

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INTERSECTIONS OF TRANSLATION OF A CLASS OF SIERPINSKI CARPETS

Fractals ◽

10.1142/s0218348x18500342 ◽

2018 ◽

Vol 26 (03) ◽

pp. 1850034

Author(s):

JIAN LU ◽

BO TAN ◽

YURU ZOU

Keyword(s):

Iterated Function System ◽

The Self ◽

Open Set Condition ◽

Moran Set ◽

Open Set ◽

Iterated Function ◽

Directed Set ◽

Self Similar ◽

Sierpinski Carpets ◽

Intersection Formula

For [Formula: see text], a middle-[Formula: see text] Sierpinski carpet [Formula: see text] is defined as the self-similar set generated by the iterated function system (IFS) [Formula: see text], where [Formula: see text] is defined by [Formula: see text] Here, [Formula: see text]. In this paper, for [Formula: see text], we investigated the equivalent characterizations of the intersection [Formula: see text] being a generalized Moran set. Furthermore, under some conditions, we show that [Formula: see text] can be represented as a graph-directed set satisfying the open set condition (OSC), and then the Hausdorff dimension can be explicitly calculated.

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Data-driven sector coupling in 5G-based smart networks

Competition and Regulation in Network Industries ◽

10.1177/1783591721992762 ◽

2021 ◽

Vol 22 (1) ◽

pp. 53-68

Author(s):

Guenter Knieps

Keyword(s):

Data Privacy ◽

Service Providers ◽

Policy Implications ◽

Network Industries ◽

5G Networks ◽

Privacy And Security ◽

Open Set ◽

Application Service ◽

Downstream Application

5G attains the role of a GPT for an open set of downstream IoT applications in various network industries and within the app economy more generally. Traditionally, sector coupling has been a rather narrow concept focusing on the horizontal synergies of urban system integration in terms of transport, energy, and waste systems, or else the creation of new intermodal markets. The transition toward 5G has fundamentally changed the framing of sector coupling in network industries by underscoring the relevance of differentiating between horizontal and vertical sector coupling. Due to the fixed mobile convergence and the large open set of complementary use cases, 5G has taken on the characteristics of a generalized purpose technology (GPT) in its role as the enabler of a large variety of smart network applications. Due to this vertical relationship, characterized by pervasiveness and innovational complementarities between upstream 5G networks and downstream application sectors, vertical sector coupling between the provider of an upstream GPT and different downstream application industries has acquired particular relevance. In contrast to horizontal sector coupling among different application sectors, the driver of vertical sector coupling is that each of the heterogeneous application sectors requires a critical input from the upstream 5G network provider and combines this with its own downstream technology. Of particular relevance for vertical sector coupling are the innovational complementarities between upstream GPT and downstream application sectors. The focus on vertical sector coupling also has important policy implications. Although the evolution of 5G networks strongly depends on the entrepreneurial, market-driven activities of broadband network operators and application service providers, the future of 5G as a GPT is heavily contingent on the role of frequency management authorities and European regulatory policy with regard to data privacy and security regulations.

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