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Stochastics and thermodynamics for equilibrium measures of holomorphic endomorphisms on complex projective spaces

Description: This article proves that the dynamical system (f,μφ) enjoys exponential decay of correlations of Hölder continuous observables as well as the Central Limit Theorem and the Law of Iterated Logarithm for the class of these variables that, in addition, satisfy a natural co-boundary condition. This article also shows that the topological pressure function t ↦ P(tφ) is real-analytic throughout the open set of all parameters t for which the potentials tφ have pressure gaps.
Date: February 14, 2014
Creator: Szostakiewicz, Michał; Urbański, Mariusz & Zdunik, Anna
Partner: UNT College of Arts and Sciences

Results in Algebraic Determinedness and an Extension of the Baire Property

Description: In this work, we concern ourselves with particular topics in Polish space theory. We first consider the space A(U) of complex-analytic functions on an open set U endowed with the usual topology of uniform convergence on compact subsets. With the operations of point-wise addition and point-wise multiplication, A(U) is a Polish ring. Inspired by L. Bers' algebraic characterization of the relation of conformality, we show that the topology on A(U) is the only Polish topology for which A(U) is a Polish ring for a large class of U. This class of U includes simply connected regions, simply connected regions excluding a relatively discrete set of points, and other domains of usual interest. One thing that we deduce from this is that, even though C has many different Polish field topologies, as long as it sits inside another Polish ring with enough complex-analytic functions, it must have its usual topology. In a different direction, we show that the bounded complex-analytic functions on the unit disk admits no Polish topology for which it is a Polish ring. We also study the Lie ring structure on A(U) which turns out to be a Polish Lie ring with the usual topology. In this case, we restrict our attention to those domains U that are connected. We extend a result of I. Amemiya to see that the Lie ring structure is determined by the conformal structure of U. In a similar vein to our ring considerations, we see that, again for certain domains U of usual interest, the Lie ring A(U) has a unique Polish topology for which it is a Polish Lie ring. Again, the Lie ring A(U) imposes topological restrictions on C. That is, C must have its usual topology when sitting inside any Polish Lie ring isomorphic to A(U). In the last ...
Date: May 2017
Creator: Caruvana, Christopher
Partner: UNT Libraries

Numerical Values of the Hausdorff and Packing Measures for Limit Sets of Iterated Function Systems

Description: 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.
Date: August 2017
Creator: Reid, James Edward
Partner: UNT Libraries

Algebraically Determined Rings of Functions

Description: Let R be any of the following rings: the smooth functions on R^2n with the Poisson bracket, the Hamiltonian vector fields on a symplectic manifold, the Lie algebra of smooth complex vector fields on C, or a variety of rings of functions (real or complex valued) over 2nd countable spaces. Then if H is any other Polish ring and φ:H →R is an algebraic isomorphism, then it is also a topological isomorphism (i.e. a homeomorphism). Moreover, many such isomorphisms between function rings induce a homeomorphism of the underlying spaces. It is also shown that there is no topology in which the ring of real analytic functions on R is a Polish ring.
Date: August 2010
Creator: McLinden, Alexander Patrick
Partner: UNT Libraries

Dynamics, Thermodynamic formalism and Perturbations of Transcendental Entire Functions of Finite Singular Type

Description: In this dissertation, we study the dynamics, fractal geometry and the topology of the Julia set of functions in the family H which is a set in the class S, the Speiser class of entire transcendental functions which have only finitely many singular values. One can think of a function from H as a generalized expanding function from the cosh family. We shall build a version of thermodynamic formalism for functions in H and we shall show among others, the existence and uniqueness of a conformal measure. Then we prove a Bowen's type formula, i.e. we show that the Hausdorff dimension of the set of returning points, is the unique zero of the pressure function. We shall also study conjugacies in the family H, perturbation of functions in the family and related dynamical properties. We define Perron-Frobenius operators for some functions naturally associated with functions in the family H and then, using fundamental properties of these operators, we shall prove the important result that the Hausdorff dimension of the subset of returning points depends analytically on the parameter taken from a small open subset of the n-dimensional parameter space.
Date: May 2005
Creator: Coiculescu, Ion
Partner: UNT Libraries

Dimension spectrum and graph directed Markov systems.

Description: In this dissertation we study graph directed Markov systems (GDMS) and limit sets associated with these systems. Given a GDMS S, by the Hausdorff dimension spectrum of S we mean the set of all positive real numbers which are the Hausdorff dimension of the limit set generated by a subsystem of S. We say that S has full Hausdorff dimension spectrum (full HD spectrum), if the dimension spectrum is the interval [0, h], where h is the Hausdorff dimension of the limit set of S. We give necessary conditions for a finitely primitive conformal GDMS to have full HD spectrum. A GDMS is said to be regular if the Hausdorff dimension of its limit set is also the zero of the topological pressure function. We show that every number in the Hausdorff dimension spectrum is the Hausdorff dimension of a regular subsystem. In the particular case of a conformal iterated function system we show that the Hausdorff dimension spectrum is compact. We introduce several new systems: the nearest integer GDMS, the Gauss-like continued fraction system, and the Renyi-like continued fraction system. We prove that these systems have full HD spectrum. A special attention is given to the backward continued fraction system that we introduce and we prove that it has full HD spectrum. This system turns out to be a parabolic iterated function system and this makes the analysis more involved. Several examples have been constructed in the past of systems not having full HD spectrum. We give an example of such a system whose limit set has positive Lebesgue measure.
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Date: May 2006
Creator: Ghenciu, Eugen Andrei
Partner: UNT Libraries

Gibbs/Equilibrium Measures for Functions of Multidimensional Shifts with Countable Alphabets

Description: Consider a multidimensional shift space with a countably infinite alphabet, which serves in mathematical physics as a classical lattice gas or lattice spin system. A new definition of a Gibbs measure is introduced for suitable real-valued functions of the configuration space, which play the physical role of specific internal energy. The variational principle is proved for a large class of functions, and then a more restrictive modulus of continuity condition is provided that guarantees a function's Gibbs measures to be a nonempty, weakly compact, convex set of measures that coincides with the set of measures obeying a form of the DLR equations (which has been adapted so as to be stated entirely in terms of specific internal energy instead of the Hamiltonians for an interaction potential). The variational equilibrium measures for a such a function are then characterized as the shift invariant Gibbs measures of finite entropy, and a condition is provided to determine if a function's Gibbs measures have infinite entropy or not. Moreover the spatially averaged limiting Gibbs measures, i.e. constructive equilibria, are shown to exist and their weakly closed convex hull is shown to coincide with the set of true variational equilibrium measures. It follows that the "pure thermodynamic phases", which correspond to the extreme points in the convex set of equilibrium measures, must be constructive equilibria. Finally, for an even smoother class of functions a method is presented to construct a compatible interaction potential and it is checked that the two different structures generate the same sets of Gibbs and equilibrium measures, respectively.
Date: May 2011
Creator: Muir, Stephen R.
Partner: UNT Libraries

Thermodynamical Formalism

Description: Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.
Date: August 2004
Creator: Chousionis, Vasileios
Partner: UNT Libraries

Conformal and Stochastic Non-Autonomous Dynamical Systems

Description: 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 ...
Date: August 2018
Creator: Atnip, Jason
Partner: UNT Libraries

Hausdorff Dimension of Shrinking-Target Sets Under Non-Autonomous Systems

Description: 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.
Date: August 2018
Creator: Lopez, Marco Antonio
Partner: UNT Libraries

Examples and Applications of Infinite Iterated Function Systems

Description: 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.
Date: August 2000
Creator: Hanus, Pawel Grzegorz
Partner: UNT Libraries

Dimensions in Random Constructions.

Description: We consider random fractals generated by random recursive constructions, prove zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.
Date: May 2002
Creator: Berlinkov, Artemi
Partner: UNT Libraries

Criticality in Cooperative Systems

Description: Cooperative behavior arises from the interactions of single units that globally produce a complex dynamics in which the system acts as a whole. As an archetype I refer to a flock of birds. As a result of cooperation the whole flock gets special abilities that the single individuals would not have if they were alone. This research work led to the discovery that the function of a flock, and more in general, that of cooperative systems, surprisingly rests on the occurrence of organizational collapses. In this study, I used cooperative systems based on self-propelled particle models (the flock models) which have been proved to be virtually equivalent to sociological network models mimicking the decision making processes (the decision making model). The critical region is an intermediate condition between a highly disordered state and a strong ordered one. At criticality the waiting times distribution density between two consecutive collapses shows an inverse power law form with an anomalous statistical behavior. The scientific evidences are based on measures of information theory, correlation in time and space, and fluctuation statistical analysis. In order to prove the benefit for a system to live at criticality, I made a flock system interact with another similar system, and then observe the information transmission for different disturbance values. I proved that at criticality the transfer of information gets the maximal efficiency. As last step, the flock model has been shown that, despite its simplicity, is sufficiently a realistic model as proved via the use of 3D simulations and computer animations.
Date: May 2012
Creator: Vanni, Fabio
Partner: UNT Libraries

Multifractal Measures

Description: The purpose of this dissertation is to introduce a natural and unifying multifractal formalism which contains the above mentioned multifractal parameters, and gives interesting results for a large class of natural measures. In Part 2 we introduce the proposed multifractal formalism and study it properties. We also show that this multifractal formalism gives natural and interesting results when applied to (nonrandom) graph directed self-similar measures in Rd and "cookie-cutter" measures in R. In Part 3 we use the multifractal formalism introduced in Part 2 to give a detailed discussion of the multifractal structure of random (and hence, as a special case, non-random) graph directed self-similar measures in R^d.
Date: May 1994
Creator: Olsen, Lars
Partner: UNT Libraries

A Collapsing Result Using the Axiom of Determinancy and the Theory of Possible Cofinalities

Description: Assuming the axiom of determinacy, we give a new proof of the strong partition relation on ω1. Further, we present a streamlined proof that J<λ+(a) (the ideal of sets which force cof Π α < λ) is generated from J<λ+(a) by adding a singleton. Combining these results with a polarized partition relation on ω1
Date: May 2001
Creator: May, Russell J.
Partner: UNT Libraries

Continuous, Nowhere-Differentiable Functions with no Finite or Infinite One-Sided Derivative Anywhere

Description: In this paper, we study continuous functions with no finite or infinite one-sided derivative anywhere. In 1925, A. S. Beskovitch published an example of such a function. Since then we call them Beskovitch functions. This construction is presented in chapter 2, The example was simple enough to clear the doubts about the existence of Besicovitch functions. In 1932, S. Saks showed that the set of Besicovitch functions is only a meager set in C[0,1]. Thus the Baire category method for showing the existence of Besicovitch functions cannot be directly applied. A. P. Morse in 1938 constructed Besicovitch functions. In 1984, Maly revived the Baire category method by finding a non-empty compact subspace of (C[0,1], || • ||) with respect to which the set of Morse-Besicovitch functions is comeager.
Date: December 1994
Creator: Lee, Jae S. (Jae Seung)
Partner: UNT Libraries

A Dynamic and Thermodynamic Approach to Complexity.

Description: The problem of establishing the correct approach to complexity is a very hot and crucial issue to which this dissertation gives some contributions. This dissertation considers two main possibilities, one, advocated by Tsallis and co-workers, setting the foundation of complexity on a generalized, non-extensive , form of thermodynamics, and another, proposed by the UNT Center for Nonlinear Science, on complexity as a new condition that, for physical systems, would be equivalent to a state of matter intermediate between dynamics and thermodynamics. In the first part of this dissertation, the concept of Kolmogorov-Sinai entropy is introduced. The Pesin theorem is generalized in the formalism of Tsallis non-extensive thermodynamics. This generalized form of Pesin theorem is used in the study of two major classes of problems, whose prototypes are given by the Manneville and the logistic map respectively. The results of these studies convince us that the approach to complexity must be made along lines different from those of the non-extensive thermodynamics. We have been convinced that the Lévy walk can be used as a prototype model of complexity, as a condition of balance between order and randomness that yields new phenomena such as aging, and multifractality. We reach the conclusions that these properties must be studied within a dynamic rather than thermodynamic perspective. The second part focuses on the study of the heart beating problem using a dynamic model, the so-called memory beyond memory, based on the Lévy walker model. It is proved that the memory beyond memory effect is more obvious in the healthy heart beating sequence. The concepts of fractal, multifractal, wavelet transformation and wavelet transform maximum modulus (WTMM) method are introduced. Artificial time sequences are generated by the memory beyond memory model to mimic the heart beating sequence. Using WTMM method, the multifratal singular spectrums of the sequences ...
Date: August 2003
Creator: Yang, Jin
Partner: UNT Libraries

The Global Structure of Iterated Function Systems

Description: 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].
Date: May 2009
Creator: Snyder, Jason Edward
Partner: UNT Libraries

Mycielski-Regular Measures

Description: Let μ be a Radon probability measure on M, the d-dimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielski-regular.
Date: August 2011
Creator: Bass, Jeremiah Joseph
Partner: UNT Libraries

Quantization Dimension for Probability Definitions

Description: 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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Date: December 2001
Creator: Lindsay, Larry J.
Partner: UNT Libraries