David Steinsaltz · The Annals of Probability · 1999
An iterated function system on $\mathscr{X}\subset\mathbb{R}^d$ is defined by successively applying an i.i.d.sequence of random Lipschitz functions from to $\mathscr{X}$ to $\mathscr{X}$. This paper shows how $F _n = f_1\circ\ldots\circ f_n$ may converge even in the absence of the strong contraction conditions, for instance, Lipschitz constant smaller than 1 on average,which earlier work has required. Instead, it is posited that there be a region of contraction which compensates for the noncontractive or even expansive part of the functions. Applications to queues, to self-modifying random walks and to random logistic maps are given.
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Alexandru Mihail, Irina Savu · Fuzzy Sets and Systems · 2023
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John W. Layman, Tad E. Womack · Computers & Graphics · 1990
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F.H. Ghane, E. Rezaali, Aliasghar Sarizadeh · Journal of Mathematical Analysis and Applications · 2018
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Huw Jones, Magnus Moar · Defense Technical Information Center (DTIC) · 2000
Iterated Function Systems (IFS) have been widely used for image compression and for generating fractal objects. Using barycentric coordinates and an extension of the concept, a range of rendering methods, such as Lambert, Gouraud and Phong shading, can he generated using IFS. By rendering a scattered collection of individual points using z-buffer and shadow buffer, the problems of clipping, hidden surface elimination and shadow generation are reduced to very simple forms The method is relatively simple to program, but will not achieve the high speeds of current sophisticated rendering methods.
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Graeme Boore · St Andrews Research Repository (St Andrews Research Repository) · 2011
This thesis concerns an active research area within fractal geometry. \nIn the first part, in Chapters 2 and 3, for directed graph iterated function systems \n(IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors \nthat cannot be the attractors of standard (1-vertex directed graph) IFSs, with \nor without separation conditions. We also calculate their exact Hausdorff measure. \nThus we are able to identify a new class of attractors for which the exact Hausdorff \nmeasure is known. \nWe give a constructive algorithm for calculating the set of gap lengths of any \nattractor as a finite union of cosets of finitely generated semigroups of positive real \nnumbers. The generators of these semigroups are contracting similarity ratios of \nsimple cycles in the directed graph. The algorithm works for any IFS defined on ℝ \nwith no limit on the number of vertices in the directed graph, provided a separation \ncondition holds. \nThe second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We \nobtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth \npacking moment of μᵤ, the self-similar measure at a vertex u, for the non-lattice case, \nwith a corresponding limit for the lattice case. We do this \n(i) for any q ∈ ℝ if the strong separation condition holds, \n(ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative \nmatrix associated with the system is irreducible. \nIn the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum \nof μᵤ to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper \nmultifractal q box-dimension with respect to μᵤ, of the set consisting of all the intersections \nof the components of Fᵤ, is strictly less than the multifractal q Hausdorff \ndimension with respect to μᵤ of Fᵤ.
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Nicholas Harland · arXiv (Cornell University) · 2012
The arithmetic function λ(n) is the exponent of the cyclic group (Z/nZ)^x. The k-th iterate of λ(n) is denoted by λk(n) In this work we will show the normal order for log(n/λk(n)) is (loglog n)k⁻¹}(logloglog n)/(k-1)! . Second, we establish a similar normal order for other iterate involving a combination of λ(n) and Φ(n). Lastly, define L(n) to be the smallest k such that λ_k(n)=1. We determine new upper and lower bounds for L(n) and conjecture a normal order.
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Pavel V. Moskalev, Alexey G. Bukhovets and Tatyana Ya. Biruchinskay · 2012
Pointwise generation and display of attractors (prefractals) of the random iterated function system (RIFS) for various combinations of probabilistic and geometric parameters of some fixed point sets (protofractals), described by Bukhovets A.G. (2012) <<a href="https://doi.org/10.1134%2FS0005117912020154" target="_top">doi:10.1134/S0005117912020154</a>>.
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Jana Bohnstengel, Marc Keßeböhmer · Journal of Functional Analysis · 2010
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Adrian Secelean · Filomat · 2011
One of the most common and most general way to generate fractals is by using iterated function systems which consists of a finite or infinitely many maps. Generalized countable iterated function systems (GCIFS) are a generalization of countable iterated function systems by considering contractions from X ? X into X instead of contractions on the metric space X to itself, where (X, d) is a compact metric space. If all contractions of a GCIFS are Lipschitz with respect to a parameter and the supremum of the Lipschitz constants is finite, then the associated attractor depends continuously on the respective parameter.
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Christiane Rousseau, Yvan Saint-Aubin · Springer undergraduate texts in mathematics and technology · 2008
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R. Selvy, P. B. Vinod Kumar · Journal of Intelligent & Fuzzy Systems · 2024
It is observed that IFSs are defined based on the concept that the iterates take only an integer number of times. This work studies the dynamics of functions, where a function can iterate r times for every r ∈ R. Utilizing concepts from fuzzy sets, r-times iterates of a function are defined for r ∈ R. The study demonstrates that the chaotic property can be generalized to this new iterative concept. The chaotic behavior of a function is then extended using this iterative concept.
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Bohnstengel, Jana, Kesseböhmer, Marc · arXiv (Cornell University) · 2009
We construct a wavelet and a generalised Fourier basis with respect to some fractal measures given by one-dimensional iterated function systems. In this paper we will not assume that these systems are given by linear contractions generalising in this way some previous work of Jorgensen and Dutkay to the non-linear setting.
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Nick Harland · arXiv (Cornell University) · 2011
The Carmichael lambda function $λ(n)$ is defined to be the smallest positive integer $m$ such that $a^m$ is congruent to 1 modulo $n,$ for all $a$ and $n$ relatively prime. The function $λ_k(n)$ is defined to be the $k$th iterate of $λ(n).$ Previous results show a normal order for $n/λ_k(n)$ where $k=1,2.$ We will show a normal order for all $k.$
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Palle E. T. Jorgensen, Keri A. Kornelson, Karen L. Shuman · 2008
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CUHK Digital Repository (Chinese University of Hong Kong) · 2002
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Fractals and dynamics in mathematics, science and the arts · 2024
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Maliheh Mohtashamipour, Alireza Zamani Bahabadi · International Journal of Bifurcation and Chaos · 2020
In the present paper, we study chaos in iterated function systems (IFS), namely dynamical systems with several generators. We introduce weak Li–Yorke chaos, chaos in branch, and weak topological chaos to perceive the role of branches to create chaos in an IFS. Moreover, we define another type of chaos, [Formula: see text]-chaos, on an IFS. Further, we find the necessary conditions to create the chaotic iterated function systems.
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Maliheh Mohtashamipour, Alireza Zamani Bahabadi · Georgian Mathematical Journal · 2019
Abstract In this paper, we define accessibility on an iterated function system (IFS) and show that it provides a sufficient condition for the transitivity of this system and its corresponding skew product. Then, by means of a certain tool, we obtain the topologically mixing property. We also give some results about the ergodicity and stability of accessibility and, further, illustrate accessibility by some examples.
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2016
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