Aimee Johnson, Kathleen Madden, Ayşe Şahı̇n · American Mathematical Society eBooks · 2017
Let {fi | 1 ≤ i ≤ n} be a finite set of n mappings from a complete metric space (E, d) onto itself; {pi | 1 ≤ i ≤ n} a discrete probability distribution, that is, a set of nonnegative real numbers such that \( \sum\nolimits_{i = 1}^n {pi} = 1 \) ; and F : E : E the mapping defined by x : F(x) = fi(x) with probability pi. The dynamical system ((E,d),F) is called an iterated function system or IFS.
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2006
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Alexander Mendez Espinel · Universidad Industrial de Santander · 2012
La propuesta de este trabajo es presentar resultados análogos al bien conocido caso de un sistema iterado de funciones con un número finito de contracciones pero generalizando el número de contracciones, el espacio en el que se definen y el tipo de noción de contracción. Las siguientes son las generalizaciones que se consideran: sistemas iterados de funciones con un número infinito de contracciones en un espacio métrico completo así como en un espacio métrico compacto, sistemas iterados de funciones con un número finito e infinito contracciones generalizadas: E-contracciones y p-contracciones. Se muestran algunos teoremas de punto fijo y las interrelaciones que existen entre as contracciones, E-contracciones y g-contracciones. Para cada sistema iterado de funciones generalizado se demuestra la existencia y la unicidad del punto fijo del operador de Hutchinson-Barnsley asociado a este. Además se prueba que los atractores de los sistemas con infinitas contracciones e infinitas contracciones generalizadas se pueden aproximar mediante los atractores de sistemas iterados de funciones finitos apropiadamente escogidos en términos de la distancia Hausdorff. Se muestran ejemplos de conjuntos que no se pueden obtener como el punto fijo de sistemas iterados de funciones con un nmñero finito de contracciones pero si cuando se consideran sistemas con un nmñero infinito de ellas.
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Michael F. Barnsley, John H. Elton, Douglas P. Hardin · Constructive Approximation · 1989
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Michael F. Barnsley, John H. Elton, Douglas P. Hardin · Constructive Approximation · 1989
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WORLD SCIENTIFIC eBooks · 2006
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Mario Peruggia · 1993
Written for researchers and developers applying Integrated Function Systems in the creation of fractal images, this book presents a modification of a widely used probabilistic algorithm for generating IFS-encoded images. The book also includes a discussion of how IFS techniques can be applied to produce animated motion pictures.
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S. M. Iacus · 2002
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R. Selvy, P. B. Vinod Kumar · Springer proceedings in mathematics & statistics · 2025
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Annina Iseli, Kevin Wildrick · Conformal Geometry and Dynamics of the American Mathematical Society · 2017
We consider a class of iterated function systems (IFSs) of contracting similarities of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"><mml:semantics><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:annotation encoding="application/x-tex">\mathbb {R}^n</mml:annotation></mml:semantics></mml:math></inline-formula>, introduced by Hutchinson, for which the invariant set possesses a natural Hölder continuous parameterization by the unit interval. When such an invariant set is homeomorphic to an interval, we give necessary conditions in terms of the similarities alone for it to possess a quasisymmetric (and as a corollary, bi-Hölder) parameterization. We also give a related necessary condition for the invariant set of such an IFS to be homeomorphic to an interval.
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Linda Keen, Nikola Lakic · 2003
We consider the iterated function system Fn = fn ◦... ◦ f2 ◦ f1 formed from the holomorphic functions in the family H(∆,Ω). The analog of stable behavior for such systems is that the limit functions be constant. We prove that a necessary and sufficient condition for stable behavior for all iterated function systems formed from H(∆,Ω) is that Ω be a proper subset of ∆. We also prove that for a given iterated function system the constant limit functions are unique if and only if Ω is relatively compact in ∆. 1
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Louisa F. Barnsley, Michael F. Barnsley, Andrew Vince · Chaos Solitons & Fractals · 2024
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Michael F. Barnsley, Louisa F. Barnsley, Andrew Vince · arXiv (Cornell University) · 2020
This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be fractals and the tiled set may be a complicated unbounded subset of $\mathbb{R}^{M}$.
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Artur Łoziński, Karol Życzkowski, Wojciech Słomczyński · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · 2003
An iterated function system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum IFS (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting, a QIFS consists of completely positive maps acting in the space of density operators. This formalism is designed to describe certain problems of nonunitary quantum dynamics. We present exemplary classical IFSs, the invariant measure of which exhibits fractal structure, and study properties of the corresponding QIFSs and their invariant states.
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Dieter van Melkebeek, Adhemar Bultheel · Lirias (KU Leuven) · 1996
In this paper, we give a theoretically founded transition from Iterated Function Systems based on affine transformations to Partitioned Iterated Function Systems. We show that there are two essential steps, namely, restricting the affine transformations, and solving the evoked problem of ink dissipation.
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P. Tharaniya, G. Jayalalitha, Pethuru Raj et al. · 2024
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P. A. Catlin · American Mathematical Monthly · 1970
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Michael Field · Springer undergraduate mathematics series · 2017
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Michael F. Barnsley, Andrew Vince · Journal of Geometric Analysis · 2011
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P. F. Duvall, L. S. Husch · Proceedings of the American Mathematical Society · 1992
In this paper, the question of which compact metric spaces can be attractors of hyperbolic iterated function systems on Euclidean space is studied. It is shown that given any finite-dimensional compact metric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there is a Cantor set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the disjoint union of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an attractor. In the process, it is proved that every such <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Lipschitz image of a Cantor set.
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