Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

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In this thesis, we consider the properties of diffeomorphisms of R3 called trace maps. We begin by introducing the definition of the trace map. The group B3 acts by trace maps on R3. The first two chapters deal with the action of a specific element of B3,called αn. In particular, we study the fixed points of αn lying on a topological subspace contained in R3, called T . We investigate the duality of the fixed points of the action ofαn, which will be defined later in the thesis.Chapter 3 involves the study of the fixed points of an element called γnm, and it generalizes the results of chapter 2. Chapter 4 involves a study of the period two points of γnm. Chapters 5-8 deal with surfaces and curves induced by trace maps, in a manner described in chapter 5. Trace maps define surfaces, and we study the intersection of those surfaces. In particular, we classify each such possible intersection.

A morphism $\sigma$ is \emph{(strongly) unambiguous} with respect to a word $\alpha$ if there is no other morphism $\tau$ that maps $\alpha$ to the same image as $\sigma$. Moreover, $\sigma$ is said to be \emph{weakly unambiguous} with respect to a word $\alpha$ if $\sigma$ is the only \emph{nonerasing} morphism that can map $\alpha$ to $\sigma(\alpha)$, i.\,e., there does not exist any other nonerasing morphism $\tau$ satisfying $\tau(\alpha) = \sigma(\alpha)$. In the first main part of the present thesis, we wish to characterise those words with respect to which there exists a weakly unambiguous \emph{length-increasing} morphism that maps a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions. \par The second main part of the present thesis studies the question of whether, for any given word, there exists a strongly unambiguous \emph{1-uniform} morphism, i.\,e., a morphism that maps every letter in the word to an image of length $1$. This problem shows some connections to previous research on \emph{fixed points} of nontrivial morphisms, i.\,e., those words $\alpha$ for which there is a morphism $\phi$ satisfying $\phi(\alpha) = \alpha$ and, for a symbol $x$ in $\alpha$, $\phi(x) \neq x$. Therefore, we can expand our examination of the existence of unambiguous morphisms to a discussion of the question of whether we can reduce the number of different symbols in a word that is not a fixed point such that the resulting word is again not a fixed point. This problem is quite similar to the setting of Billaud's Conjecture, the correctness of which we prove for a special case.

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

Doctor of Philosophy / Department of Mathematics / Pietro Poggi-Corradini / We consider iteration of an analytic self-map f of the unit ball in the N-dimensional complex space C[superscript]N. Many facts were established about such maps and their dynamics in the 1-dimensional case (i.e. for self-maps of the unit disk), and we generalize some of them in higher dimensions. In one dimension, the classical Denjoy-Wolff theorem states the convergence of forward iterates to a unique attracting fixed point, while backward iterates have much more complicated nature. However, under additional conditions (when the hyperbolic distance between two consecutive points stays bounded), backward iteration sequence converges to a point on the boundary of the unit disk, which happens to be a fixed point with multiplier greater than or equal to 1. In this paper, we explore backward-iteration sequences in higher dimension. Our main result shows that in the case when f is hyperbolic or elliptic, such sequences with bounded hyperbolic step converge to a point on the boundary, other than the Denjoy-Wolff (attracting) point. These points are called boundary repelling fixed points (BRFPs) and possess several nice properties. In particular, in the case when such points are isolated from other BRFPs, they are completely characterized as limits of backward iteration sequences. Similarly to classical results, it is also possible to construct a (semi) conjugation to an automorphism of the unit ball. However, unlike in the 1-dimensional case, not all BRFPs are isolated, and we present several counterexamples to show that. We conclude with some results in the parabolic case.

O conhecido Teorema de Poincaré afirma: O número de rotação de homeomorfismo do círculo S^1 que preserva orientação é racional se, e somente se, o homeomorfismo possui um ponto periódico cujo período é igual ao denominador de tal racional. Na presente dissertação estudamos resultados análogos, ao resultado acima mencionado, para homeomorfismos do anel A=S^1 x I homotópicos à identidade. Mais precisamente, estudaremos o famoso Teorema de Poincaré - Birkhoff e algumas versões devidas a J. Franks. Isto será feito impondo algumas condições no conjunto de rotação, o qual é uma generalização do número de rotação para homeomorfismos do círculo. / The well known Poincaré's Theorem state: The rotation number of an orientation preserving circle homeomorphism is rational if, only if, the homeomorphism has a periodic point of period equal to denominator of the rational. In this monograph we study results analogous, to the result above mentioned, for homeomorphisms of A=S^1 x I homotophics to the identity. More precisely, we study the famous Poincaré - Birkhoff Theorem and some versions obtained by J. Franks. This it will be done imposing some conditions in the rotation set, which is generalization of the rotation number for circle homeomorphisms.

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Rearrangements are two measurable real-valued functions that have equal measure of pre-images of upper level sets. In this thesis, I will investigate several matters and problems relating to rearrangements: the relationship between assumptions on the measure space and desirable properties of the set of rearrangements, and the validity of rearrangement inequalities; generalising the Mountain Pass Lemma over rearrangements; and applying topological degree theory to boundary value problems involving rearrangements. From suppositions on the measure space, such as the measure space having finite measure and no atoms, it can proved that the set of rearrangements is contractible and locally contractible. The Mountain Pass Lemma over rearrangements can be generalised, so instead of considering continuous paths from the closed unit interval to the set of rearrangements; it will consider the continuous functions from the closed unit disc into the set of rearrangements. Topological degree theory is used to associate admissible triples of functions, sets and points with integers. These methods will be applied to a boundary value problem involving rearrangements, where the domain is almost equal to the union of balls, which has been studied using variational methods, providing new multiplicity results. The minimum number of solutions to this boundary value problem is found to be related exponentially to the number of balls contained in the domain.

O conhecido Teorema de Poincaré afirma: O número de rotação de homeomorfismo do círculo S^1 que preserva orientação é racional se, e somente se, o homeomorfismo possui um ponto periódico cujo período é igual ao denominador de tal racional. Na presente dissertação estudamos resultados análogos, ao resultado acima mencionado, para homeomorfismos do anel A=S^1 x I homotópicos à identidade. Mais precisamente, estudaremos o famoso Teorema de Poincaré - Birkhoff e algumas versões devidas a J. Franks. Isto será feito impondo algumas condições no conjunto de rotação, o qual é uma generalização do número de rotação para homeomorfismos do círculo. / The well known Poincaré's Theorem state: The rotation number of an orientation preserving circle homeomorphism is rational if, only if, the homeomorphism has a periodic point of period equal to denominator of the rational. In this monograph we study results analogous, to the result above mentioned, for homeomorphisms of A=S^1 x I homotophics to the identity. More precisely, we study the famous Poincaré - Birkhoff Theorem and some versions obtained by J. Franks. This it will be done imposing some conditions in the rotation set, which is generalization of the rotation number for circle homeomorphisms.