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Gadde, Erland. "Stable iterated function systems." Doctoral thesis, Umeå universitet, Institutionen för matematik, teknik och naturvetenskap, 1992. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-100370.
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The purpose of this thesis is to generalize the growing theory of iterated function systems (IFSs). Earlier, hyperbolic IFSs with finitely many functions have been studied extensively. Also, hyperbolic IFSs with infinitely many functions have been studied. In this thesis, more general IFSs are studied. The Hausdorff pseudometric is studied. This is a generalization of the Hausdorff metric. Wide and narrow limit sets are studied. These are two types of limits of sequences of sets in a complete pseudometric space. Stable Iterated Function Systems, a kind of generalization of hyperbolic IFSs, are defined. Some different, but closely related, types of stability for the IFSs are considered. It is proved that the IFSs with the most general type of stability have unique attractors. Also, invariant sets, addressing, and periodic points for stable IFSs are studied. Hutchinson’s metric (also called Vaserhstein’s metric) is generalized from being defined on a space of probability measures, into a class of norms, the £-norms, on a space of real measures (on certain metric spaces). Under rather general conditions, it is proved that these norms, when they are restricted to positive measures, give rise to complete metric spaces with the metric topology coinciding with the weak*-topology. Then, IFSs with probabilities (IFSPs) are studied, in particular, stable IFSPs. The £-norm-results are used to prove that, as in the case of hyperbolic IFSPs, IFSPs with the most general kind of stability have unique invariant measures. These measures are ”attractive”. Also, an invariant measure is constructed by first ”lifting” the IFSP to the code space. Finally, it is proved that the Random Iteration Algorithm in a sense will ”work” for some stable IFSPs.Diss. Umeå : Umeå universitet, 1992
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Reid, James Edward. "Numerical Values of the Hausdorff and Packing Measures for Limit Sets of Iterated Function Systems." Thesis, University of North Texas, 2017. https://digital.library.unt.edu/ark:/67531/metadc1011825/.
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In the context of fractal geometry, the natural extension of volume in Euclidean space is given by Hausdorff and packing measures. These measures arise naturally in the context of iterated function systems (IFS). For example, if the IFS is finite and conformal, then the Hausdorff and packing dimensions of the limit sets agree and the corresponding Hausdorff and packing measures are positive and finite. Moreover, the map which takes the IFS to its dimension is continuous. Developing on previous work, we show that the map which takes a finite conformal IFS to the numerical value of its packing measure is continuous. In the context of self-similar sets, we introduce the super separation condition. We then combine this condition with known density theorems to get a better handle on finding balls of maximum density. This allows us to extend the work of others and give exact formulas for the numerical value of packing measure for classes of Cantor sets, Sierpinski N-gons, and Sierpinski simplexes.
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Alexander, Simon. "Multiscale Methods in Image Modelling and Image Processing." Thesis, University of Waterloo, 2005. http://hdl.handle.net/10012/1179.
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The field of modelling and processing of 'images' has fairly recently become important, even crucial, to areas of science, medicine, and engineering. The inevitable explosion of imaging modalities and approaches stemming from this fact has become a rich source of mathematical applications. 'Imaging' is quite broad, and suffers somewhat from this broadness. The general question of 'what is an image?' or perhaps 'what is a natural image?' turns out to be difficult to address. To make real headway one may need to strongly constrain the class of images being considered, as will be done in part of this thesis. On the other hand there are general principles that can guide research in many areas. One such principle considered is the assertion that (classes of) images have multiscale relationships, whether at a pixel level, between features, or other variants. There are both practical (in terms of computational complexity) and more philosophical reasons (mimicking the human visual system, for example) that suggest looking at such methods. Looking at scaling relationships may also have the advantage of opening a problem up to many mathematical tools.
This thesis will detail two investigations into multiscale relationships, in quite different areas. One will involve Iterated Function Systems (IFS), and the other a stochastic approach to reconstruction of binary images (binary phase descriptions of porous media). The use of IFS in this context, which has often been called 'fractal image coding', has been primarily viewed as an image compression technique. We will re-visit this approach, proposing it as a more general tool. Some study of the implications of that idea will be presented, along with applications inferred by the results. In the area of reconstruction of binary porous media, a novel, multiscale, hierarchical annealing approach is proposed and investigated.
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Reis, Glauco dos Santos. "Uma abordagem de compressão de imagens através de sistemas de funções iteradas." Universidade Presbiteriana Mackenzie, 2011. http://tede.mackenzie.br/jspui/handle/tede/1408.
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Previous issue date: 2011-08-22Fundo Mackenzie de Pesquisa
A new image compression technique is proposed, based on the affine transformations (ATs) that define an iterated function system (IFS). Previous related research in the field has shown that an image may be approximated by iteratively subjecting a set of sub-regions to a group of ATs. In this case, the original image should be partitioned in regions, and each one of the active pixels are transformed by the AT. The new transformed set should be approximated to other image regions. This iterated execution to find ATs for the best set of areas might result in smaller storage space since the similar areas might be replaced by AT coefficients. Despite this advantage, the technique is computationally intensive, because both the sub-regions and the corresponding ATs that have to be searched for. Here, a new form of similarity is proposed, based on the successive points generated by the iteration of affine transformations. By understanding an AT as a discrete dynamical system, with each image point represented by an iteration of the AT, the method captures similarities between these points, namely, those with the same color in the image; by saving the starting point and the transformations coefficients, the points can be iterated back, to reconstruct the original image. This results in lighter computational effort, since the comparison is made point by point, instead of region by region. Experiments were made on a group of 10 images, representing a broad set of distinct features and resolutions. The proposed algorithm competes in terms of storage size, when compared to JPEG, mainly when the image size is small, and the number of colors are reduced, as currently happens for most images used in the Internet. Although the proposed method is faster than the traditional method for IFS compression, it is slower than common file formats like JPEG.
Uma nova técnica para compressão de imagens é proposta, baseada em conjuntos de transformações afins (affine transformations - ATs), normalmente conhecidos como sistemas de funções iteradas (iterated function system -IFS). Pesquisas anteriores mostraram que uma imagem poderia ser aproximada pela aplicação de um grupo de ATs em conjuntos de sub-regiões da imagem, de forma iterativa. Através deste processo, a imagem original seria subdividida em regiões e sobre a coordenada de cada ponto habilitado de cada região seria aplicada uma transformação afim. O resultado representaria um novo conjunto de pontos similares a outras regiões da imagem. A execução de forma iterada deste processo de identificação das ATs para o maior conjunto de regiões similares de uma determinada imagem permitiria uma redução no armazenamento, já que as regiões similares poderiam ser armazenadas como os coeficientes das transformações afins. Apesar desta vantagem em termos de compressão, a técnica é computacionalmente intensiva, pela busca exaustiva de sub-regiões e das ATs geradoras, de forma a proporcionar o melhor preenchimento em outras regiões da imagem. Esta pesquisa propõe uma nova forma de compressão baseada em ATs, utilizando a sequência de pontos gerada pela iteração das ATs. Entendendo uma AT como um sistema dinâmico em tempo discreto, cada novo ponto identificado é consequência direta da iteração da AT sobre o ponto anterior, permitindo a captura de similaridades nesta sequência de pontos. Através do salvamento dos coeficientes das ATs e das coordenadas iniciais, é possível a reconstrução da imagem pela iteração da AT a partir do ponto inicial. Isto pode resultar em menor esforço computacional, pois apenas comparações simples de pontos são necessárias, ao invés de comparações entre os pontos de regiões da imagem. Foram feitos experimentos em um conjunto de 10 classes de imagens, representando um espectro de diferentes características gerais e resoluções. O algoritmo proposto rivaliza em termos de armazenamento quando comparado ao formato JPEG, principalmente para imagens de pequeno tamanho e com número de cores reduzidas, como as utilizadas com frequência na Internet. Apesar de ser mais rápido para a compressão do que outros métodos baseados em IFS, ele é mais lento do que métodos clássicos como o JPEG.
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Boore, Graeme C. "Directed graph iterated function systems." Thesis, University of St Andrews, 2011. http://hdl.handle.net/10023/2109.
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This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μ[subscript(u)], the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μ[subscript(u)], of the set consisting of all the intersections of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff dimension with respect to μ[subscript(u)] of F[subscript(u)].
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Vines, Greg. "Signal modeling with iterated function systems." Diss., Georgia Institute of Technology, 1993. http://hdl.handle.net/1853/13315.
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Hardin, Douglas Patten. "Hyperbolic iterated function systems and applications." Diss., Georgia Institute of Technology, 1985. http://hdl.handle.net/1853/30864.
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Giles, Paul A. "Iterated function systems and shape representation." Thesis, Durham University, 1990. http://etheses.dur.ac.uk/6188/.
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We propose the use of iterated function systems as an isomorphic shape representation scheme for use in a machine vision environment. A concise description of the basic theory and salient characteristics of iterated function systems is presented and from this we develop a formal framework within which to embed a representation scheme. Concentrating on the problem of obtaining automatically generated two-dimensional encodings we describe implementations of two solutions. The first is based on a deterministic algorithm and makes simplifying assumptions which limit its range of applicability. The second employs a novel formulation of a genetic algorithm and is intended to function with general data input. Keywords: Machine Vision, Shape Representation, Iterated Function Systems, Genetic Algorithms.
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Kieninger, Bernd. "Iterated function systems on compact Hausdorff spaces /." Aachen : Shaker, 2002. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=010050648&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.
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Paul, Baldine-Brunel. "Video Compression based on iterated function systems." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/13553.
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Massopust, Peter Robert. "Space curves generated by iterated function systems." Diss., Georgia Institute of Technology, 1986. http://hdl.handle.net/1853/29339.
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Snyder, Jason Edward. "The Global Structure of Iterated Function Systems." Thesis, University of North Texas, 2009. https://digital.library.unt.edu/ark:/67531/metadc9917/.
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I study sets of attractors and non-attractors of finite iterated function systems. I provide examples of compact sets which are attractors of iterated function systems as well as compact sets which are not attractors of any iterated function system. I show that the set of all attractors is a dense Fs set and the space of all non-attractors is a dense Gd set it the space of all non-empty compact subsets of a space X. I also investigate the small trans-finite inductive dimension of the space of all attractors of iterated function systems generated by similarity maps on [0,1].
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Chiu, Anthony. "Iterated function systems that contract on average." Thesis, University of Manchester, 2015. https://www.research.manchester.ac.uk/portal/en/theses/iterated-function-systems-that-contract-on-average(38f391bf-142f-4fcd-9144-fbe5ca8ec7ba).html.
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Consider an iterated function system (IFS) that does not necessarily contract uniformly, but instead contracts on average after a finite number of iterations. Under some technical assumptions, previous work by Barnsley, Demko, Elton and Geronimo has shown that such an IFS has a unique invariant probability measure, whilst many (such as Peigné, Hennion and Hervé, Guivarc'h and le Page, Santos and Walkden) have shown that (for different function spaces) the transfer operator associated with the IFS is quasi-compact. A result due to Keller and Liverani allows one to deduce whether the transfer operator remains quasi-compact under suitable, small perturbations. The first part of this thesis proves a large deviations result for IFSs that contract on average using skew product transfer operators, a technique used by Broise to prove a similar result for dynamical systems. The remaining chapters introduce a notion of 'coupled IFSs', an analogue of the traditional coupled map lattices where the base, unperturbed behaviour is determined by an underlying dynamical system. We use transfer operators and Keller and Liverani's theorem to prove that quasi-compactness of the transfer operator is preserved for 'product IFSs' under small perturbations and for coupled IFSs. This allows us to prove a central limit theorem with a rate of convergence for the coupled IFS.
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Snyder, Jason Edward Urbaʹnski Mariusz. "The global structure of iterated function systems." [Denton, Tex.] : University of North Texas, 2009. http://digital.library.unt.edu/permalink/meta-dc-9917.
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Tino, Peter, and Georg Dorffner. "Recurrent neural networks with iterated function systems dynamics." SFB Adaptive Information Systems and Modelling in Economics and Management Science, WU Vienna University of Economics and Business, 1998. http://epub.wu.ac.at/948/1/document.pdf.
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We suggest a recurrent neural network (RNN) model with a recurrent part corresponding to iterative function systems (IFS) introduced by Barnsley [1] as a fractal image compression mechanism. The key idea is that 1) in our model we avoid learning the RNN state part by having non-trainable connections between the context and recurrent layers (this makes the training process less problematic and faster), 2) the RNN state part codes the information processing states in the symbolic input stream in a well-organized and intuitively appealing way. We show that there is a direct correspondence between the Rényi entropy spectra characterizing the input stream and the spectra of Renyi generalized dimensions of activations inside the RNN state space. We test both the new RNN model with IFS dynamics and its conventional counterpart with trainable recurrent part on two chaotic symbolic sequences. In our experiments, RNNs with IFS dynamics outperform the conventional RNNs with respect to information theoretic measures computed on the training and model generated sequences. (author's abstract)Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"
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Hanus, Pawel Grzegorz. "Examples and Applications of Infinite Iterated Function Systems." Thesis, University of North Texas, 2000. https://digital.library.unt.edu/ark:/67531/metadc2642/.
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The aim of this work is the study of infinite conformal iterated function systems. More specifically, we investigate some properties of a limit set J associated to such system, its Hausdorff and packing measure and Hausdorff dimension. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We use the concept of scaling functions to obtain some result about 1-dimensional systems. We discuss particular examples of infinite iterated function systems derived from complex continued fraction expansions with restricted entries. Each system is obtained from an infinite number of contractions. We show that under certain conditions the limit sets of such systems possess zero Hausdorff measure and positive finite packing measure. We include an algorithm for an approximation of the Hausdorff dimension of limit sets. One numerical result is presented. In this thesis we also explore the concept of positively recurrent function. We use iterated function systems to construct a natural, wide class of such functions that have strong ergodic properties.
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Reuter, Laurie Hodges. "Rendering and magnification of fractals using iterated function systems." Diss., Georgia Institute of Technology, 1987. http://hdl.handle.net/1853/8250.
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Peña, Helena [Verfasser]. "Affine Iterated Function Systems, invariant measures and their approximation / Helena Peña." Greifswald : Universitätsbibliothek Greifswald, 2017. http://d-nb.info/113081551X/34.
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Qi, Xiaomin. "Fixed points, fractals, iterated function systems and generalized support vector machines." Licentiate thesis, Mälardalens högskola, Utbildningsvetenskap och Matematik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-33511.
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In this thesis, fixed point theory is used to construct a fractal type sets and to solve data classification problem. Fixed point method, which is a beautiful mixture of analysis, topology, and geometry has been revealed as a very powerful and important tool in the study of nonlinear phenomena. The existence of fixed points is therefore of paramount importance in several areas of mathematics and other sciences. In particular, fixed points techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory and physics. In Chapter 2 of this thesis it is demonstrated how to define and construct a fractal type sets with the help of iterations of a finite family of generalized F-contraction mappings, a class of mappings more general than contraction mappings, defined in the context of b-metric space. This leads to a variety of results for iterated function system satisfying a different set of contractive conditions. The results unify, generalize and extend various results in the existing literature. In Chapter 3, the theory of support vector machine for linear and nonlinear classification of data and the notion of generalized support vector machine is considered. In the thesis it is also shown that the problem of generalized support vector machine can be considered in the framework of generalized variation inequalities and results on the existence of solutions are established.FUSION
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Bernat, Andrew. "Which partition scheme for what image?, partitioned iterated function systems for fractal image compression." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2002. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/MQ65602.pdf.
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Bodin, Mats. "Characterisations of function spaces on fractals." Doctoral thesis, Umeå : Department of Mathematics and Mathematical Statistics, Umeå University, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-580.
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Horn, Alastair N. "Iterated function systems, the parallel progressive synthesis of fractal tiling structures and their applications to computer graphics." Thesis, University of Oxford, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.257907.
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Godland, Philipp [Verfasser], and Gerold [Akademischer Betreuer] Alsmeyer. "Markov renewal theory in the analysis of random strings and iterated function systems / Philipp Godland ; Betreuer: Gerold Alsmeyer." Münster : Universitäts- und Landesbibliothek Münster, 2020. http://d-nb.info/1213805252/34.
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Lopez, Marco Antonio. "Hausdorff Dimension of Shrinking-Target Sets Under Non-Autonomous Systems." Thesis, University of North Texas, 2018. https://digital.library.unt.edu/ark:/67531/metadc1248505/.
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For a dynamical system on a metric space a shrinking-target set consists of those points whose orbit hit a given ball of shrinking radius infinitely often. Historically such sets originate in Diophantine approximation, in which case they describe the set of well-approximable numbers. One aspect of such sets that is often studied is their Hausdorff dimension. We will show that an analogue of Bowen's dimension formula holds for such sets when they are generated by conformal non-autonomous iterated function systems satisfying some natural assumptions.
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Troscheit, Sascha. "Dimension theory of random self-similar and self-affine constructions." Thesis, University of St Andrews, 2017. http://hdl.handle.net/10023/11033.
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This thesis is structured as follows. Chapter 1 introduces fractal sets before recalling basic mathematical concepts from dynamical systems, measure theory, dimension theory and probability theory. In Chapter 2 we give an overview of both deterministic and stochastic sets obtained from iterated function systems. We summarise classical results and set most of the basic notation. This is followed by the introduction of random graph directed systems in Chapter 3, based on the single authored paper [T1] to be published in Journal of Fractal Geometry. We prove that these attractors have equal Hausdorff and upper box-counting dimension irrespective of overlaps. It follows that the same holds for the classical models introduced in Chapter 2. This chapter also contains results about the Assouad dimensions for these random sets. Chapter 4 is based on the single authored paper [T2] and establishes the box-counting dimension for random box-like self-affine sets using some of the results and the notation developed in Chapter 3. We give some examples to illustrate the results. In Chapter 5 we consider the Hausdorff and packing measure of random attractors and show that for reasonable random systems the Hausdorff measure is zero almost surely. We further establish bounds on the gauge functions necessary to obtain positive or finite Hausdorff measure for random homogeneous systems. Chapter 6 is based on a joint article with J. M. Fraser and J.-J. Miao [FMT] to appear in Ergodic Theory and Dynamical Systems. It is chronologically the first and contains results that were extended in the paper on which Chapter 3 is based. However, we will give some simpler, alternative proofs in this section and crucially also find the Assouad dimension of some random self-affine carpets and show that the Assouad dimension is always `maximal' in both measure theoretic and topological meanings.
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Atnip, Jason. "Conformal and Stochastic Non-Autonomous Dynamical Systems." Thesis, University of North Texas, 2018. https://digital.library.unt.edu/ark:/67531/metadc1248519/.
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In this dissertation we focus on the application of thermodynamic formalism to non-autonomous and random dynamical systems. Specifically we use the thermodynamic formalism to investigate the dimension of various fractal constructions via the, now standard, technique of Bowen which he developed in his 1979 paper on quasi-Fuchsian groups. Bowen showed, roughly speaking, that the dimension of a fractal is equal to the zero of the relevant topological pressure function. We generalize the results of Rempe-Gillen and Urbanski on non-autonomous iterated function systems to the setting of non-autonomous graph directed Markov systems and then show that the Hausdorff dimension of the fractal limit set is equal to the zero of the associated pressure function provided the size of the alphabets at each time step do not grow too quickly. In trying to remove these growth restrictions, we present several other systems for which Bowen's formula holds, most notably ascending systems.
We then use these various constructions to investigate the Hausdorff dimension of various subsets of the Julia set for different large classes of transcendental meromorphic functions of finite order which have been perturbed non-autonomously. In particular we find lower and upper bounds for the dimension of the subset of the Julia set whose points escape to infinity, and in many cases we find the exact dimension. While the upper bound was known previously in the autonomous case, the lower bound was not known in this setting, and all of these results are new in the non-autonomous setting.
We also use transfer operator techniques to prove an almost sure invariance principle for random dynamical systems for which the thermodynamical formalism has been well established. In particular, we see that if a system exhibits a fiberwise spectral gap property and the base dynamical system is sufficiently well behaved, i.e. it exhibits an exponential decay of correlations, then the almost sure invariance principle holds. We then apply these results to uniformly expanding random systems like those studied by Mayer, Skorulski, and Urbanski and Denker and Gordin.
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Lindsay, Larry J. "Quantization Dimension for Probability Definitions." Thesis, University of North Texas, 2001. https://digital.library.unt.edu/ark:/67531/metadc3008/.
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The term quantization refers to the process of estimating a given probability by a discrete probability supported on a finite set. The quantization dimension Dr of a probability is related to the asymptotic rate at which the expected distance (raised to the rth power) to the support of the quantized version of the probability goes to zero as the size of the support is allowed to go to infinity. This assumes that the quantized versions are in some sense ``optimal'' in that the expected distances have been minimized. In this dissertation we give a short history of quantization as well as some basic facts. We develop a generalized framework for the quantization dimension which extends the current theory to include a wider range of probability measures. This framework uses the theory of thermodynamic formalism and the multifractal spectrum. It is shown that at least in certain cases the quantization dimension function D(r)=Dr is a transform of the temperature function b(q), which is already known to be the Legendre transform of the multifractal spectrum f(a). Hence, these ideas are all closely related and it would be expected that progress in one area could lead to new results in another. It would also be expected that the results in this dissertation would extend to all probabilities for which a quantization dimension function exists. The cases considered here include probabilities generated by conformal iterated function systems (and include self-similar probabilities) and also probabilities generated by graph directed systems, which further generalize the idea of an iterated function system.
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Ross, Emily L. "Examination of traditional and v-variable fractals." Honors in the Major Thesis, University of Central Florida, 2009. http://digital.library.ucf.edu/cdm/ref/collection/ETH/id/1318.
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This item is only available in print in the UCF Libraries. If this is your Honors Thesis, you can help us make it available online for use by researchers around the world by following the instructions on the distribution consent form at http://library.ucf.edu/Systems/DigitalInitiatives/DigitalCollections/InternetDistributionConsentAgreementForm.pdf You may also contact the project coordinator, Kerri Bottorff, at kerri.bottorff@ucf.edu for more information.Bachelors
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Wilson, Deborah Ann Stoffer. "A Study of the Behavior of Chaos Automata." Kent State University / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=kent1478955376070686.
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De, Castro Gilles. "C*-algèbres associées à certains systèmes dynamiques et leurs états KMS." Phd thesis, Université d'Orléans, 2009. http://tel.archives-ouvertes.fr/tel-00541042.
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D'abord, on étudie trois façons d'associer une C*-algèbre à une transformation continue. Ensuite, nousdonnons une nouvelle définition de l'entropie. Nous trouvons des relations entre les états KMS des algèbrespréalablement définies et les états d'équilibre, donné par un principe variationnel. Dans la seconde partie,nous étudions les algèbres de Kajiwara-Watatani associées à un système des fonctions itérées. Nouscomparons ces algèbres avec l'algèbre de Cuntz et le produit croisé. Enfin, nous étudions les états KMS desalgèbres de Kajiwara-Watatani pour les actions provenant d'un potentiel et nous trouvouns des relationsentre ces états et les mesures trouvée dans une version de le théorème de Ruelle-Perron-Frobenius pour lessystèmes de fonctions itérées.
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Castro, Gilles Gonçalves de. "C*-álgebras associadas a certas dinâmicas e seus estados KMS." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2009. http://hdl.handle.net/10183/18824.
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D'abord, on étudie trois façons d'associer une C*-algèbre à une transformation continue. Ensuite, nous donnons une nouvelle définition de l'entropie. Nous trouvons des relations entre les états KMS des algèbres préalablement définies et les états d'équilibre, donné par un principe variationnel. Dans la seconde partie, nous étudions les algèbres de Kajiwara-Watatani associees a un système des fonctions itérées. Nous comparons ces algèbres avec l'algèbre de Cuntz et le produit croisé. Enfin, nous étudions les états KMS des algèbres de Kajiwara-Watatani pour les actions provenant d'un potentiel et nous trouvouns des relations entre ces états et les mesures trouvee dans une version de le théorème de Ruelle-Perron-Frobenius pour les systèmes de fonctions itérées.Primeiramente, estudamos três formas de associar uma C*-álgebra a uma transformação contínua. Em seguida, damos uma nova definição de entropia. Relacionamos, então, os estados KMS das álgebras anteriormente definidas com os estados de equilibro, vindos de um princípio variacional. Na segunda parte, estudamos as álgebras de Kajiwara-Watatani associadas a um sistema de funções iteradas. Comparamos tais álgebras com a álgebra de Cuntz e a álgebra do produto cruzado. Finalmente, estudamos os estados KMS das álgebras de Kajiwara-Watatani para ações vindas de um potencial e relacionamos tais estados KMS com medidas encontradas numa versão do teorema de Ruelle-Perron-Frobenius para sistemas de funções iteradas.
First, we study three ways of associating a C*-algebra to a continuous map. Then, we give a new de nition of entropy. We relate the KMS states of the previously de ned algebras with the equilibrium states, given by a variational principle. In the second part, we study the Kajiwara-Watatani algebras associated to iterated function system. We compare these algebras with the Cuntz algebra and the crossed product. Finally, we study the KMS states of the Kajiwara-Watatani algebras for actions coming from a potential and we relate such states with measures found in a version of the Ruelle-Perron- Frobenius theorem for iterated function systems.
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Pesee, Chatchai. "Stochastic modelling of financial processes with memory and semi-heavy tails." Thesis, Queensland University of Technology, 2005. https://eprints.qut.edu.au/16057/2/Chatchai%20Pesee%20Thesis.pdf.
Full textAbstract:
This PhD thesis aims to study financial processes which have semi-heavy-tailed marginal distributions and may exhibit memory. The traditional Black-Scholes model is expanded to incorporate memory via an integral operator, resulting in a class of market models which still preserve the completeness and arbitragefree conditions needed for replication of contingent claims. This approach is used to estimate the implied volatility of the resulting model. The first part of the thesis investigates the semi-heavy-tailed behaviour of financial processes. We treat these processes as continuous-time random walks characterised by a transition probability density governed by a fractional Riesz- Bessel equation. This equation extends the Feller fractional heat equation which generates a-stable processes. These latter processes have heavy tails, while those processes generated by the fractional Riesz-Bessel equation have semi-heavy tails, which are more suitable to model financial data. We propose a quasi-likelihood method to estimate the parameters of the fractional Riesz- Bessel equation based on the empirical characteristic function. The second part considers a dynamic model of complete financial markets in which the prices of European calls and puts are given by the Black-Scholes formula. The model has memory and can distinguish between historical volatility and implied volatility. A new method is then provided to estimate the implied volatility from the model. The third part of the thesis considers the problem of classification of financial markets using high-frequency data. The classification is based on the measure representation of high-frequency data, which is then modelled as a recurrent iterated function system. The new methodology developed is applied to some stock prices, stock indices, foreign exchange rates and other financial time series of some major markets. In particular, the models and techniques are used to analyse the SET index, the SET50 index and the MAI index of the Stock Exchange of Thailand.
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Pesee, Chatchai. "Stochastic Modelling of Financial Processes with Memory and Semi-Heavy Tails." Queensland University of Technology, 2005. http://eprints.qut.edu.au/16057/.
Full textAbstract:
This PhD thesis aims to study financial processes which have semi-heavy-tailed marginal distributions and may exhibit memory. The traditional Black-Scholes model is expanded to incorporate memory via an integral operator, resulting in a class of market models which still preserve the completeness and arbitragefree conditions needed for replication of contingent claims. This approach is used to estimate the implied volatility of the resulting model. The first part of the thesis investigates the semi-heavy-tailed behaviour of financial processes. We treat these processes as continuous-time random walks characterised by a transition probability density governed by a fractional Riesz- Bessel equation. This equation extends the Feller fractional heat equation which generates a-stable processes. These latter processes have heavy tails, while those processes generated by the fractional Riesz-Bessel equation have semi-heavy tails, which are more suitable to model financial data. We propose a quasi-likelihood method to estimate the parameters of the fractional Riesz- Bessel equation based on the empirical characteristic function. The second part considers a dynamic model of complete financial markets in which the prices of European calls and puts are given by the Black-Scholes formula. The model has memory and can distinguish between historical volatility and implied volatility. A new method is then provided to estimate the implied volatility from the model. The third part of the thesis considers the problem of classification of financial markets using high-frequency data. The classification is based on the measure representation of high-frequency data, which is then modelled as a recurrent iterated function system. The new methodology developed is applied to some stock prices, stock indices, foreign exchange rates and other financial time series of some major markets. In particular, the models and techniques are used to analyse the SET index, the SET50 index and the MAI index of the Stock Exchange of Thailand.
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Vestin, Albin, and Gustav Strandberg. "Evaluation of Target Tracking Using Multiple Sensors and Non-Causal Algorithms." Thesis, Linköpings universitet, Reglerteknik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-160020.
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Today, the main research field for the automotive industry is to find solutions for active safety. In order to perceive the surrounding environment, tracking nearby traffic objects plays an important role. Validation of the tracking performance is often done in staged traffic scenarios, where additional sensors, mounted on the vehicles, are used to obtain their true positions and velocities. The difficulty of evaluating the tracking performance complicates its development. An alternative approach studied in this thesis, is to record sequences and use non-causal algorithms, such as smoothing, instead of filtering to estimate the true target states. With this method, validation data for online, causal, target tracking algorithms can be obtained for all traffic scenarios without the need of extra sensors. We investigate how non-causal algorithms affects the target tracking performance using multiple sensors and dynamic models of different complexity. This is done to evaluate real-time methods against estimates obtained from non-causal filtering. Two different measurement units, a monocular camera and a LIDAR sensor, and two dynamic models are evaluated and compared using both causal and non-causal methods. The system is tested in two single object scenarios where ground truth is available and in three multi object scenarios without ground truth. Results from the two single object scenarios shows that tracking using only a monocular camera performs poorly since it is unable to measure the distance to objects. Here, a complementary LIDAR sensor improves the tracking performance significantly. The dynamic models are shown to have a small impact on the tracking performance, while the non-causal application gives a distinct improvement when tracking objects at large distances. Since the sequence can be reversed, the non-causal estimates are propagated from more certain states when the target is closer to the ego vehicle. For multiple object tracking, we find that correct associations between measurements and tracks are crucial for improving the tracking performance with non-causal algorithms.
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Ramkumar, M. "Some New Methods For Improved Fractal Image Compression." Thesis, 1996. https://etd.iisc.ac.in/handle/2005/1897.
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Ramkumar, M. "Some New Methods For Improved Fractal Image Compression." Thesis, 1996. http://etd.iisc.ernet.in/handle/2005/1897.
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"Iterated function systems and multifractals." 2002. http://library.cuhk.edu.hk/record=b6073402.
Full textAbstract:
by Wang Xiang-Yang."May 2002."
Thesis (Ph.D.)--Chinese University of Hong Kong, 2002.
Includes bibliographical references (p. 95-99).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Mode of access: World Wide Web.
Abstracts in English and Chinese.
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Chu, Hsueh-Ting, and 朱學亭. "Applications of Fractals With Iterated Function Systems." Thesis, 2002. http://ndltd.ncl.edu.tw/handle/82080071881042303096.
Full textAbstract:
博士國立清華大學
資訊工程學系
91
Fractals with Iterated Function Systems (IFSs) can be used in two applications: object modeling and image compression. However, there are problems on both of the applications to be solved. For modeling 2D or 3D graphic objects, we point out two significant problems that deter the ongoing development of fractals. The first problem is where fractal objects are. It is necessary to compute the bounding extent of a fractal object before the rendering procedure. Unfortunately, the current bounding extents were determined by trial and error methods. We develop an algorithm to compute tight bounding boxes of fractal objects. The second problem is how to determine the intersections of rays and fractal objects. The intersection problem is a prerequisite of rendering 3D objects with ray tracing. We develop another algorithm, called stencil tracing, to compute the intersections of rays and fractal objects. Based on the solutions, we are able to draw 2D fractal pictures at any resolution and to render fractal objects in synthetic 3D scenes. On the other hand, we tackle the obstacles of fractal image compression. Fractal image compression is notorious for its very slow encoding procedure. Thus we suggest a new coding scheme to accelerate the encoding procedure. In fractal image compression, an image is divided into overlapped domain blocks and non-overlapped range blocks. For each range block, the mapped domain block that is affine-similar to the range block is determined. The collection of the affine mappings forms a Partitioned Iterated Function System (PIFS). The time consuming problem comes from massive computation needed for searching a best match among a multitude of range-domain block pairs. In the past, people exploited block classification algorithms or fast greedy searching algorithms to solve the problem. We give another approach. We try to build indices onto the domain image. Only those range and domain blocks with same index keys are compared. The domain indexing technology is different from variant domain classification ones because we don’t cluster all of the domain blocks really. Thus we can accelerate the encoding procedure on a large scale. We consider the efficiency of the decoding procedure in our new scheme, too. We investigate the kernel of a partitioned iterated function system that must be computed iteratively. Thus it also benefits from the speed-up of decoding images.
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Korfanty, Emily Rose. "Étale equivalence relations and C*-algebras for iterated function systems." Thesis, 2020. http://hdl.handle.net/1828/12486.
Full textAbstract:
There is a long history of interesting connections between topological dynamical systems and C*-algebras. Iterated function systems are an important topic in dynamics, but the diversity of these systems makes it challenging to develop an associated class of C*-algebras. Kajiwara and Watatani were the first to construct a C*-algebra from an iterated function system. They used an algebraic approach involving Cuntz-Pimsner algebras; however, when investigating properties such as ideal structure, they needed to assume that the functions in the system are the inverse branches of a continuous map. This excludes many famous examples, such as the standard functions used to construct the Siérpinski Gasket. In this thesis, we provide a construction of an inductive limit of étale equivalence relations for a broad class of affine iterated function systems, including the Siérpinski Gasket and its relatives, and consider the associated C*-algebras. This approach provides a more dynamical perspective, leading to interesting results that emphasize how properties of the dynamics appear in the C*-algebras. In particular, we show that the C*-algebras are isomorphic for conjugate systems, and find ideals related to the open set condition. In the case of the Siérpinski Gasket, we find explicit isomorphisms to subalgebras of the continuous functions from the attractor to a matrix algebra. Finally, we consider the K-theory of the inductive limit of these algebras.Graduate
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Lim, Chuan Chin, and 淩聖欽. "Evoked Potential Analysis of Rat Motor Cortex Based on Iterated Function Systems." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/v7338q.
Full textAbstract:
碩士國立臺北科技大學
電腦與通訊研究所
102
Brain research has been study for more than 40years and brain neurons signals is required for understanding brain functions. Motor cortex is the main part that controls actions. Therefore, recording and analyzing the brain neurons signals in the motor cortex will help to understand relationships between actions and brain neurons signals. In the study, we use multichannel electrode of micro-wire to obtain the neural signals from motor cortex of rats. Different kind of actions signals was classified and applied ICA for obtaining independent source signals. IFS were used to analyze relationships between neural signals and actions. Assume that an action is merged by neural signals that combined from different path and neuron signals of the nervous systems that produced by motor cortex, then we encode by inverse IFS . Different kind of actions is encoded by IFS. Analyze the results to understand the relationships between the neural signals and actions.
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Silvestri, Stefano. "The Dynamics of Semigroups of Contraction Similarities on the Plane." Thesis, 2019. http://hdl.handle.net/1805/19909.
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Indiana University-Purdue University Indianapolis (IUPUI)Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M. Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.
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Demers, Matthew. "Fractal Imaging Theory and Applications beyond Compression." Thesis, 2012. http://hdl.handle.net/10214/3634.
Full textAbstract:
The use of fractal-based methods in imaging was first popularized with fractal
image compression in the early 1990s. In this application, one seeks to approximate
a given target image by the fixed point of a contractive operator called the fractal
transform. Typically, one uses Local Iterated Function Systems with Grey-Level
Maps (LIFSM), where the involved functions map a parent (domain) block in an
image to a smaller child (range) block and the grey-level maps adjust the shading
of the shrunken block. The fractal transform is defined by the collection of optimal
parent-child pairings and parameters defining the grey-level maps. Iteration of the
fractal transform on any initial image produces an approximation of the fixed point
and, hence, an approximation of the target image. Since the parameters defining
the LIFSM take less space to store than the target image does, image compression is
achieved.This thesis extends the theoretical and practical frameworks of fractal imaging to
one involving a particular type of multifunction that captures the idea that there are
typically many near-optimal parent-child pairings. Using this extended machinery, we
treat three application areas. After discussing established edge detection methods,
we present a fractal-based approach to edge detection with results that compare
favourably to the Sobel edge detector. Next, we discuss two methods of information
hiding: first, we explore compositions of fractal transforms and cycles of images
and apply these concepts to image-hiding; second, we propose and demonstrate an
algorithm that allows us to securely embed with redundancy a binary string within
an image. Finally, we discuss some theory of certain random fractal transforms with
potential applications to texturing.The Natural Sciences and Engineering Research Council and the University of Guelph helped to provide financial support for this research.
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(6983546), Stefano Silvestri. "The Dynamics of Semigroups of Contraction Similarities on the Plane." Thesis, 2019.
Find full textAbstract:
Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M.
Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.
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Szczepanek, Anna. "Quantum dynamical entropy of unitary operators in finite-dimensional state spaces." Praca doktorska, 2020. https://ruj.uj.edu.pl/xmlui/handle/item/277285.
Full textAbstract:
Kwantowa entropia dynamiczna jest miara nieredukowalnej losowości wyników następujących po sobie pomiarów układu kwantowego, który pomiędzy dwoma kolejnymi pomiarami podlega ewolucji unitarnej. Łączna ewolucja takiego układu opisywana jest za pomocą częściowego iterowanego układu funkcyjnego (PIFS, partial iterated function system) na zespolonej przestrzeni rzutowej, zaś entropia dynamiczna wyraża się za pomocą tzw. wzoru całkowego Blackwella. Uzyskanie efektywnego wzoru na entropie dynamiczna nie sprawia trudności jedynie w sytuacji, gdy pomiar składa się z operatorów o rzędzie jeden. W pierwszej części rozprawy badamy entropie dynamiczna względem pomiarów rzutowych (PVM; projection valued measure), które zawierają operatory o rzędzie większym niż jeden. Najprostszym przypadkiem tego problemu jest układ trójwymiarowy oraz pomiar złożony z dwóch rzutowań (o rzędach jeden i dwa). Przypadek ten był badany przez Crutchfielda i Wiesner dla pewnego konkretnego operatora unitarnego oraz dwóch pomiarów. W rozprawie podajemy klasyfikacje wszystkich tego typu układów trójwymiarowych: wyróżniamy osiem typów łańcuchów Markowa, które mogą zostać wygenerowane w przestrzeni stanów kwantowych, po czym dla każdego typu wyprowadzamy efektywny wzór na entropie dynamiczna oraz entropie nadwyżkowa. W drugiej części rozważamy pojecie kwantowej entropii dynamicznej niezależnej od pomiaru. Wielkość ta jest zdefiniowana jako maksimum kwantowej entropii dynamicznej (względem pomiaru) po klasie pomiarów złożonych z rzutowań jednowymiarowych (rank-1 PVM), którym odpowiadają bazy ortonormalne przestrzeni zespolonej. Wyróżniamy klasę operatorów chaotycznych, czyli operatorów o maksymalnej entropii. Korzystając z pojęcia zespolonej macierzy Hadamarda, podajemy warunek konieczny na chaotyczność, który wyraża się w języku śladu i wyznacznika operatora. Dla wymiarów dwa i trzy ten warunek jest również wystarczający, co pozwala na obliczenie objętości zbioru operatorów chaotycznych w grupie unitarnej (względem miary Haara), a także, w wymiarze dwa, średniej wartości kwantowej entropii dynamicznej. Uzyskujemy także efektywny warunek konieczny (w języku śladu) na chaotyczność operatora w wymiarze piec. Układy wyżej wymiarowe rozważane są w sytuacji wielu kubitów (układów dwuwymiarowych), z których każdy podlega lokalnej ewolucji opisywanej tym samym operatorem unitarnym. Chaotyczność tego operatora okazuje się być warunkiem koniecznym i wystarczającym osiągania przez cały układ maksymalnej entropii dynamicznej. Drugim analizowanym przypadkiem jest układ złożony z jednego kubitu o nietrywialnej ewolucji unitarnej oraz towarzyszących mu kubitów pomocniczych (o ewolucji trywialnej). Pokazujemy, ze takie rozszerzanie układu o pomocnicze kubity prowadzi do spadku entropii dynamicznej (unormowanej liczba kubitów) oraz wyznaczamy granice entropii dynamicznej przy liczbie kubitów pomocniczych dążącej do nieskończoności - granica ta okazuje się być zawsze dodatnia. Wyróżniamy także klasę uporczywie chaotycznych operatorów unitarnych jako tych, które są chaotyczne pomimo obecności dowolnie wielu pomocniczych kubitów, oraz podajemy charakteryzacje operatorów uporczywie chaotycznych w wymiarach dwa i trzy. Kwantowa entropie dynamiczna niezależna od pomiaru można zdefiniować także jako supremum entropii dynamicznej względem pomiaru po szerszej klasie tzw. pomiarów uogólnionych, którym odpowiadają miary półspektralne (POVM; positive operator valued measure). Wykazujemy, ze w wymiarze dwa maksymalizacja (odpowiednio znormalizowanej) entropii po zbiorach: mniejszym, rank-1 PVM-ów, i większym, rank-1 POVM-ów, daje ten sam wynik.Quantum dynamical entropy measures the irreducible randomness in the sequences of outcomes generated by a repetitively measured quantum system that between two consecutive measurements is subject to unitary evolution. Joint evolution of such a system can be described by a Partial Iterated Function System (PIFS) acting on a complex projective space and its dynamical entropy is given by the so-called Blackwell integral formula. It is only in the case of measurements consisting exclusively of rank-1 operators that a closed-form expression for quantum dynamical entropy can be fairly easily derived. In the first part of the dissertation we study quantum dynamical entropy with respect to projective measurements (PVMs) that contain projectors of ranks higher than one. The simplest case of this problem is a three-dimensional quantum system measured with a PVM consisting of two projectors, one of which has rank two and the other has rank one. This case was investigated by Crutchfield & Wiesner, who analysed a specific unitary operator with respect to two measurements. We provide a classification of all such three-dimensional systems, identifying eight types of Markov chains that they can generate in the space of quantum states and then deriving for each chain type efficient formulae for quantum dynamical entropy and excess entropy. In the second part we consider the notion of quantum dynamical entropy independent of measurement. This quantity is defined as the maximum of quantum dynamical entropy (with respect to a measurement) over the class of rank-1 PVMs, which correspond to orthonormal bases of the underlying complex space. We distinguish the class of chaotic unitary operators as those unitaries that attain the maximal value of quantum dynamical entropy. Employing the notion of a complex Hadamard matrix, we derive a necessary condition for chaoticity in terms of the trace and determinant of a unitary. In dimensions two and three this condition is sufficient as well, which allows for the calculation of the volume of chaotic unitaries in the unitary ensemble (with respect to the Haar measure) and, in dimension two, the average value of quantum dynamical entropy. We also obtain an effective necessary trace condition for chaoticity in dimension five. Higher dimensional systems are studied in the particular case of several qubits that are all subject to the (local) action of the same unitary. We prove that this unitary is chaotic if and only if the dynamical entropy of the composite system is maximal. We also analyse the case of non-trivial unitary dynamics affecting one qubit, while the remaining ancillary qubits are governed by the trivial dynamics. We show that quantum dynamical entropy (per one qubit) eventually decreases (i.e., when sufficiently many ancillary qubits have been taken into account) and we compute its limit as the number of ancillary qubits grows to infinity - this limit turns out to be strictly positive. Furthermore, we distinguish the class of stubbornly chaotic unitaries as those that are chaotic despite the presence of arbitrarily many ancillary qubits. We establish a necessary trace condition for a unitary to be stubbornly chaotic and characterise stubbornly chaotic unitaries in dimensions two and three. Quantum dynamical entropy independent of measurement can be also defined as the supremum of quantum dynamical entropy with respect to a measurement over the bigger class of generalised measurements, which correspond to semispectral measures (POVM; positive operator valued measure). We prove that in dimension two the maximisation over the bigger set of rank-1 POVMs leads to the same result as the maximisation over the smaller set of rank-1 PVMs.