In the first chapter; we consider the invariant measure mu generated by an integral self-affine IFS. We prove that any integral self-affine measure with a common contracting matrix can be expressed as a vector-valued self-affine measure with an IFS satisfying the open set condition (OSC). The same idea can also be applied to scaling functions of refinement equations, we extended a well known necessary and sufficient condition for the existence of L1-solutions of lattice refinement equations. We then apply this vector-valued form to study the integral self-affine sets, we obtain an algorithm for the Lebesgue measure of integral self-affine region and an algorithm for the Hausdorff dimension of a class of self-affine sets. The vector-value setup also provides an easy way to consider the L q-spectrum and the multifractal formulism for self-similar measures. As an application we can conclude the differentiability of the Lq spectrum (for q > 0) of any integral self-similar measure with a common contracting matrix. / In this thesis, we study the invariant measures and sets generated by iterated function systems (IFS). The systems have been extensively studied in the frame work of Hutchinson [Hut]. For the iteration, it is often assumed that the IFS satisfies the open set condition (OSC), a non-overlap condition in the iteration. One of the advantage of the OSC is that the point in K can be uniquely represented in a symbolic space except for a mu-zero set and many important results have been obtained. Our special interest in this thesis is to transform an invariant measure with overlaps to a vector-valued form with non overlaps. The advantage of this vector-valued form is that locally the measure can be expressed as a product of matrices. / The problem considered in the third chapter is on the choice of the invariant open set in the finite type condition (FTC). From definition, the FTC depends on the choice of the invariant open set. We show that, in one dimensional case, if the IFS satisfies the FTC for some invariant open interval then it satisfies the FTC with all invariant open sets. To our surprising, we find a counter-example to show that, in high dimensional case, the invariant open set can not be chosen arbitrarily even if the IFS satisfies the OSC and generates a tile. / The second chapter is devoted to the absolute continuity of self-affine (real-valued or vector-valued) measures and some properties of the boundary of the invariant set. For self-similar IFS with a common contracting ratio, there is a necessary and sufficient condition for the self-similar measure to be absolutely continuous with respect to the Lebesgue measure (under the weak separation condition (WSC)). In our consideration we first extend the definition of WSC to self-affine IFS. Then we generalize the previous condition to obtain a necessary and sufficient condition for the self-affine vector-valued measures to be absolutely continuous with respect to the Lebesgue measure. As an application, we prove that the boundary of all integral self-affine set has zero Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding invariant (real-valued or vector-valued) measure is supported either in V or in ∂ V. / by Deng Qirong. / "March 2005." / Adviser: Ka-Sing Lau. / Source: Dissertation Abstracts International, Volume: 67-01, Section: B, page: 0301. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 87-91). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract in English and Chinese. / School code: 1307.

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.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2009. / Title from title screen (site viewed January 5, 2010). PDF text: viii, 65 p. : ill. ; 507 K. UMI publication number: AAT 3360158. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (M. Sc.)--Dept. of Mathematics and Statistics, Concordia University, 1996. / "May 1996." Includes bibliographical references. Available also on the Internet.

We study dynamical systems in the complex plane under the effect of constant noise. We show for a wide class of polynomial equations that the ergodic property is valid in the associated stochastic perturbation if and only if the noise added is in the direction transversal to all unstable trajectories of the deterministic system. This has the interpretation that noise in the "right" direction prevents the process from being unstable: a fundamental, but not well-understood, geometric principle which seems to underlie many other similar equations. The result is proven by using Lyapunov functions and geometric control theory.

[pt] Alguns difeomorfismos de superfícies fechadas possuem apenas medidas invariantes triviais, isto é, medidas cujo suporte está contido no conjunto de pontos fixos. Resultados dessa natureza fazem uso fundamental da classificação dos homeomorfismos de superfície, tornando-os típicos da dimensão 2. Nós atacamos esse problema mostrando que difeomorfismos de superfícies que admitem medidas invariantes não-triviais exibem uma forma de crescimento linear positivo. As técnicas utilizadas são elementares e uma parte significativa dos resultados continua válida em dimensões mais altas. / [en] Some diffeomorphisms of closed surfaces only have trivial invariant probabilities, i.e., those supported on the set of fixed points. Results of this nature make extensive use of the classification of surface homeomorphisms, making them typical of dimension 2. We attack this problem by showing that surface diffeomorphisms admiting non-trivial invariant probabilities exhibit some sort of positive linear growth. The techniques used are elementary and a significant part of the results remains valid in higher dimensions.

QC 20100414

This thesis contributes towards the maximal-in-time well-posedness theory of three nonlinear dispersive partial differential equations (PDEs). We are interested in questions that extend beyond the usual well-posedness theory: what is the ultimate fate of solutions? How does Hamiltonian structure influence PDE dynamics? How does randomness, within the PDE or the initial data, interact with well-posedness of the Cauchy problem? The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of surface water waves. We prove a mass concentration property for H^1-solutions, analogous to the one known for the L^2-critical nonlinear Schrodinger equation. We also prove a mass concentration result for L^2-solutions. The second topic of this thesis is the invariance of the Gibbs measure for the (gauge transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure supported on H^s for s<1/2, and local solutions to the quartic KdV cannot be obtained below H^{1/2} by using the standard fixed point method. We exhibit nonlinear smoothing when the initial data are randomized, and establish almost sure local well-posedness for the (gauge transformed) quartic KdV below H^{1/2}. Then, using the invariance of the Gibbs measure for the finite-dimensional system of ODEs given by projection onto the first N>0 modes of the trigonometric basis, we extend the local solutions of the (gauge transformed) quartic KdV to global solutions, and prove the invariance of the Gibbs measure under the flow. Inverting the gauge, we establish almost sure global well-posedness of the (ungauged) periodic quartic KdV below H^{1/2}. The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equation. This equation is studied as a toy model for the stochastic Burgers equation, which appears in the description of a randomly growing interface. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This thesis provides a result in this direction: after smoothing the additive noise (by a fractional derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.

This thesis contributes towards the maximal-in-time well-posedness theory of three nonlinear dispersive partial differential equations (PDEs). We are interested in questions that extend beyond the usual well-posedness theory: what is the ultimate fate of solutions? How does Hamiltonian structure influence PDE dynamics? How does randomness, within the PDE or the initial data, interact with well-posedness of the Cauchy problem? The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic Davey-Stewartson system, which appears in the description of surface water waves. We prove a mass concentration property for H^1-solutions, analogous to the one known for the L^2-critical nonlinear Schrodinger equation. We also prove a mass concentration result for L^2-solutions. The second topic of this thesis is the invariance of the Gibbs measure for the (gauge transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure supported on H^s for s<1/2, and local solutions to the quartic KdV cannot be obtained below H^{1/2} by using the standard fixed point method. We exhibit nonlinear smoothing when the initial data are randomized, and establish almost sure local well-posedness for the (gauge transformed) quartic KdV below H^{1/2}. Then, using the invariance of the Gibbs measure for the finite-dimensional system of ODEs given by projection onto the first N>0 modes of the trigonometric basis, we extend the local solutions of the (gauge transformed) quartic KdV to global solutions, and prove the invariance of the Gibbs measure under the flow. Inverting the gauge, we establish almost sure global well-posedness of the (ungauged) periodic quartic KdV below H^{1/2}. The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equation. This equation is studied as a toy model for the stochastic Burgers equation, which appears in the description of a randomly growing interface. We are interested in rigorously proving the invariance of white noise for the stochastic KdV-Burgers equation. This thesis provides a result in this direction: after smoothing the additive noise (by a fractional derivative), we establish (almost sure) local well-posedness of the stochastic KdV-Burgers equation with white noise as initial data. We also prove a global well-posedness result under an additional smoothing of the noise.

We review the concept of invariant measure and study conditions under which linear combinations of averages along periodic orbits are dense in the space of invariant measures. / Revisamos el concepto de medida invariante y estudiamos condiciones bajo las cuales combinaciones lineales de promedios a lo largo de órbitas periódicas son densas en el espacio de medidas invariantes.