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Alexandru Mihail - One of the best experts on this subject based on the ideXlab platform.
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a new algorithm that generates the image of the attractor of a generalized Iterated Function System
Numerical Algorithms, 2020Co-Authors: Radu Miculescu, Alexandru Mihail, Silviu UrziceanuAbstract:We provide a new algorithm (called the grid algorithm) designed to generate the image of the attractor of a generalized Iterated Function System on a finite dimensional space and we compare it with the deterministic algorithm regarding generalized Iterated Function Systems presented by Jaros et al. (Numer. Algorithms 73, 477–499, 2016).
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A new algorithm that generates the image of the attractor of a generalized Iterated Function System
arXiv: Dynamical Systems, 2019Co-Authors: Radu Miculescu, Alexandru Mihail, Silviu UrziceanuAbstract:We provide a new algorithm (called the grid algorithm) designed to generate the image of the attractor of a generalized Iterated Function System on a finite dimensional space and we compare it with the deterministic algorithm regarding generalized Iterated Function Systems presented by P. Jaros, L. Maslanka and F. Strobin in [Algorithms generating images of attractors of generalized Iterated Function Systems, Numer. Algorithms, 73 (2016), 477-499].
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Iterated Function Systems Consisting of $$\varvec{\varphi }$$-Contractions
Results in Mathematics, 2017Co-Authors: Loredana Ioana, Alexandru MihailAbstract:In present times, there has been a considerable effort to generalize the classical notion of Iterated Function System. We’ll present in this paper Iterated Function Systems on a compact metric space consisting of \(\varphi \)-contractions and prove that such an Iterated Function System necessarily has an associated fractal set and an associated fractal measure.
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Iterated Function Systems Consisting of $$\varvec{\varphi }$$ φ -Contractions
Results in Mathematics, 2017Co-Authors: Loredana Ioana, Alexandru MihailAbstract:In present times, there has been a considerable effort to generalize the classical notion of Iterated Function System. We’ll present in this paper Iterated Function Systems on a compact metric space consisting of $$\varphi $$ -contractions and prove that such an Iterated Function System necessarily has an associated fractal set and an associated fractal measure.
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Iterated Function Systems consisting of phi-max-contractions have attractor
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Flavian Georgescu, Radu Miculescu, Alexandru MihailAbstract:We associate to each Iterated Function System consisting of phi-max-contractions an operator (on the space of continuous Functions from the shift space on the metric space corresponding to the System) having a unique fixed point whose image turns out to be the attractor of the System. Moreover, we prove that the unique fixed point of the operator associated to an Iterated Function System consisting of convex contractions is the canonical projection from the shift space on the attractor of the System.
Radu Miculescu - One of the best experts on this subject based on the ideXlab platform.
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a new algorithm that generates the image of the attractor of a generalized Iterated Function System
Numerical Algorithms, 2020Co-Authors: Radu Miculescu, Alexandru Mihail, Silviu UrziceanuAbstract:We provide a new algorithm (called the grid algorithm) designed to generate the image of the attractor of a generalized Iterated Function System on a finite dimensional space and we compare it with the deterministic algorithm regarding generalized Iterated Function Systems presented by Jaros et al. (Numer. Algorithms 73, 477–499, 2016).
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A new algorithm that generates the image of the attractor of a generalized Iterated Function System
arXiv: Dynamical Systems, 2019Co-Authors: Radu Miculescu, Alexandru Mihail, Silviu UrziceanuAbstract:We provide a new algorithm (called the grid algorithm) designed to generate the image of the attractor of a generalized Iterated Function System on a finite dimensional space and we compare it with the deterministic algorithm regarding generalized Iterated Function Systems presented by P. Jaros, L. Maslanka and F. Strobin in [Algorithms generating images of attractors of generalized Iterated Function Systems, Numer. Algorithms, 73 (2016), 477-499].
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The canonical projection associated to certain possibly infinite generalized Iterated Function System as a fixed point
arXiv: Classical Analysis and ODEs, 2018Co-Authors: Radu Miculescu, Silviu UrziceanuAbstract:In this paper, influenced by the ideas from A. Mihail, The canonical projection between the shift space of an IIFS and its attractor as a fixed point, Fixed Point Theory Appl., 2015, Paper No. 75, 15 p., we associate to every generalized Iterated Function System F (of order m) an operator H defined on C^m and taking values on C, where C stands for the space of continuous Functions from the shift space on the metric space corresponding to the System. We provide sufficient conditions (on the constitutive Functions of F) for the operator H to be continuous, contraction, phi-contraction, Meir-Keeler or contractive. We also give sufficient condition under which H has a unique fixed point. Moreover, we prove that, under these circumstances, the closer of the imagine of the fixed point is the attractor of F and that the fixed point is the canonical projection associated to F. In this way we give a partial answer to the open problem raised on the last paragraph of the above mentioned Mihail's paper.
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Iterated Function Systems consisting of phi-max-contractions have attractor
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Flavian Georgescu, Radu Miculescu, Alexandru MihailAbstract:We associate to each Iterated Function System consisting of phi-max-contractions an operator (on the space of continuous Functions from the shift space on the metric space corresponding to the System) having a unique fixed point whose image turns out to be the attractor of the System. Moreover, we prove that the unique fixed point of the operator associated to an Iterated Function System consisting of convex contractions is the canonical projection from the shift space on the attractor of the System.
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A generalization of Istratescu's fixed point theorem for convex contractions
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Radu Miculescu, Alexandru MihailAbstract:In this paper we prove a generalization of Istr\u{a}\c{t}escu's theorem for convex contractions. More precisely, we introduce the concept of Iterated Function System consisting of convex contractions and prove the existence and uniqueness of the attractor of such a System. In addition we study the properties of the canonical projection from the code space into the attractor of an Iterated Function System consisting of convex contractions.
Loredana Ioana - One of the best experts on this subject based on the ideXlab platform.
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Iterated Function Systems Consisting of $$\varvec{\varphi }$$-Contractions
Results in Mathematics, 2017Co-Authors: Loredana Ioana, Alexandru MihailAbstract:In present times, there has been a considerable effort to generalize the classical notion of Iterated Function System. We’ll present in this paper Iterated Function Systems on a compact metric space consisting of \(\varphi \)-contractions and prove that such an Iterated Function System necessarily has an associated fractal set and an associated fractal measure.
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Iterated Function Systems Consisting of $$\varvec{\varphi }$$ φ -Contractions
Results in Mathematics, 2017Co-Authors: Loredana Ioana, Alexandru MihailAbstract:In present times, there has been a considerable effort to generalize the classical notion of Iterated Function System. We’ll present in this paper Iterated Function Systems on a compact metric space consisting of $$\varphi $$ -contractions and prove that such an Iterated Function System necessarily has an associated fractal set and an associated fractal measure.
Trubee Davison - One of the best experts on this subject based on the ideXlab platform.
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A Positive Operator-Valued Measure for an Iterated Function System
Acta Applicandae Mathematicae, 2018Co-Authors: Trubee DavisonAbstract:Given an Iterated Function System (IFS) on a complete and separable metric space Y $Y$ , there exists a unique compact subset X ⊆ Y $X \subseteq Y$ satisfying a fixed point relation with respect to the IFS. This subset is called the attractor set, or fractal set, associated to the IFS. The attractor set supports a specific Borel probability measure, called the Hutchinson measure, which itself satisfies a fixed point relation. P. Jorgensen generalized the Hutchinson measure to a projection-valued measure, under the assumption that the IFS does not have essential overlap (Jorgensen in Adv. Appl. Math. 34(3):561–590, 2005 ; Operator Theory, Operator Algebras, and Applications, pp. 13–26, 2006 ). In previous work, we developed an alternative approach to proving the existence of this projection-valued measure (Davison in Acta Appl. Math. 140(1):11–22, 2015 ; Acta Appl. Math. 140(1):23–25, 2015 ; Generalizing the Kantorovich metric to projection-valued measures: with an application to Iterated Function Systems, 2015 ). The situation when the IFS exhibits essential overlap has been studied by Jorgensen and colleagues in Jorgenson et al. (J. Math. Phys. 48(8):083511, 35, 2007 ). We build off their work to generalize the Hutchinson measure to a positive-operator valued measure for an arbitrary IFS, that may exhibit essential overlap. This work hinges on using a generalized Kantorovich metric to define a distance between positive operator-valued measures. It is noteworthy to mention that this generalized metric, which we use in our previous work as well, was also introduced by R.F. Werner to study the position and momentum observables, which are central objects of study in the area of quantum theory (Werner in J. Quantum Inf. Comput. 4(6):546–562, 2004 ). We conclude with a discussion of Naimark’s dilation theorem with respect to this positive operator-valued measure, and at the beginning of the paper, we prove a metric space completion result regarding the classical Kantorovich metric.
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a positive operator valued measure for an Iterated Function System
arXiv: Functional Analysis, 2016Co-Authors: Trubee DavisonAbstract:Given an Iterated Function System (IFS) on a complete and separable metric space $Y$, there exists a unique compact subset $X \subseteq Y$ satisfying a fixed point relation with respect to the IFS. This subset is called the attractor set, or fractal set, associated to the IFS. The attractor set supports a specific Borel probability measure, called the Hutchinson measure, which itself satisfies a fixed point relation. P. Jorgensen generalized the Hutchinson measure to a projection-valued measure, under the assumption that the IFS does not have essential overlap. In previous work, we developed an alternative approach to proving the existence of this projection-valued measure. The situation when the IFS exhibits essential overlap has been studied by Jorgensen and colleagues in. We build off their work to generalize the Hutchinson measure to a positive-operator valued measure for an arbitrary IFS, that may exhibit essential overlap. This work hinges on using a generalized Kantorovich metric to define a distance between positive operator-valued measures. It is noteworthy to mention that this generalized metric, which we use in our previous work as well, was also introduced by R.F. Werner to study the position and momentum observables, which are central objects of study in the area of quantum theory. We conclude with a discussion of Naimark's dilation theorem with respect to this positive operator-valued measure, and at the beginning of the paper, we prove a metric space completion result regarding the classical Kantorovich metric.
Michael F Barnsley - One of the best experts on this subject based on the ideXlab platform.
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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, 2014Co-Authors: Michael F Barnsley, Krzysztof LeśniakAbstract: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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the conley attractors of an Iterated Function System
Bulletin of The Australian Mathematical Society, 2013Co-Authors: Michael F Barnsley, Andrew VinceAbstract:We investigate the topological and metric properties of attractors of an Iterated Function System (IFS) whose Functions may not be contractive. We focus, in particular, on invertible IFSs of nitely many maps on a com-pact metric space. We rely on ideas of Kieninger and McGehee and Wiandt, restricted to what is, in many ways, a simpler setting, but focused on a special type of attractor, namely point-fibred invariant sets. This allows us to give short proofs of some of the key ideas. 10.1017/S0004972713000348
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Fractal homeomorphism for bi-affine Iterated Function Systems
International Journal of Applied Nonlinear Science, 2013Co-Authors: Michael F Barnsley, Andrew VinceAbstract:The paper concerns fractal homeomorphism between the attractors of two bi-affine Iterated Function Systems. After a general discussion of bi-affine Functions, conditions are provided under which a bi-affine Iterated Function System is contractive, thus guaranteeing an attractor. After a general discussion of fractal homeomorphism, fractal homeomorphisms are constructed for a specific type of bi-affine Iterated Function System.
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the chaos game on a general Iterated Function System from a topological point of view
arXiv: Dynamical Systems, 2012Co-Authors: Michael F Barnsley, Krzysztof LeśniakAbstract: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 generation the orbit. A Baire category counterpart of almost sure convergence is presented; and a link between topological and probabilistic methods is observed.
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The chaos game on a general Iterated Function System
Ergodic Theory and Dynamical Systems, 2010Co-Authors: Michael F Barnsley, Andrew VinceAbstract:The main theorem of this paper establishes conditions under which the ‘chaos game’ algorithm almost surely yields the attractor of an Iterated Function System. The theorem holds in a very general setting, even for non-contractive Iterated Function Systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.
