Strong mixing measures and invariant sets in linear dynamics

The Ph.D. Thesis “Strong mixing measures and invariant sets in linear dynamics” has three differenced parts. Chapter 0 introduces the notation, definitions and the basic results that will be needed troughout the thesis. There is a first part consisting of Chapters 1 and 2, where we study the relation between the Frequent Hypercyclicity Criterion and the existence of strongly-mixing Borel probability measures. A third chapter, where we focus our attention on frequent hypercyclicity for translation C0-semigroups, and the last part corresponding to Chapters 4 and 5, where we study dynamical properties satisfied by autonomous and non-autonomous linear dynamical systems on certain invariant sets. In what follows, we give a brief description of each chapter: In Chapter 1, we construct strongly mixing Borel probability T-invariant measures with full support for operators on F-spaces which satisfy the Frequent Hypercyclicity Criterion. Moreover, we provide examples of operators that verify this criterion and we also show that this result can be improved in the case of chaotic unilateral backward shifts. The contents of this chapter have been published in [88] and [12]. In Chapter 2, we show that the Frequent Hypercyclicity Criterion for C0- semigroups, which was given by Mangino and Peris in [82], ensures the existence of invariant strongly mixing measures with full support. We will provide several examples, that range from birth-and-death models to the Black-Scholes equation, which illustrate these results. All the results of this chapter have been published in [86]. In Chapter 3, we focus our attention on one of the most important tests C0-semigroups, the translation semigroup. Inspired in the work of Bayart and Ruzsa in [22], where they characterize frequent hypercyclicity of weighted backward shifts we characterize frequently hypercyclic translation C0-semigroups on C ρ 0 (R) and L ρ p(R). Moreover, we first review some known results on the dynamics of the translation C0-semigroups. Later we state and prove a characterization of frequent hypercyclicity for weighted pseudo shifts in terms of the weights that will be used later to obtain a characterization of frequent hypercyclicity for translation C0-semigroups on C ρ 0 (R). Finally we study the case of L ρ p(R). We will also establish an analogy between the study of frequent hypercyclicity for the translation C0-semigroup in L ρ p(R) and the corresponding one for backward shifts on weighted sequence spaces. The contents of this chapter have been included in [81]. Chapter 4 is devoted to study hypercyclicity, Devaney chaos, topological mixing properties and strong mixing in the measure-theoretic sense for operators on topological vector spaces with invariant sets. More precisely, we establish links between the fact of satisfying any of our dynamical properties on certain invariant sets, and the corresponding property on the closed linear span of the invariant set, or on the union of the invariant sets. Viceversa, we give conditions on the operator (or C0-semigroup) to ensure that, when restricted to the invariant set, it satisfies certain dynamical property. Particular attention is given to the case of positive operators and semigroups on lattices, and the (invariant) positive cone. The contents of this chapter have been published in [85]. In the last chapter, motivated by the work of Balibrea and Oprocha [4], where they obtained several results about weak mixing and chaos for nonautonomous discrete systems on compact sets, we study mixing properties for nonautonomous linear dynamical systems that are induced by the corresponding dynamics on certain invariant sets. All the results of this chapter have been published in [87].Murillo Arcila, M. (2015). Strong mixing measures and invariant sets in linear dynamics [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48519TESI

Climate dynamics and fluid mechanics: Natural variability and related uncertainties

The purpose of this review-and-research paper is twofold: (i) to review the role played in climate dynamics by fluid-dynamical models; and (ii) to contribute to the understanding and reduction of the uncertainties in future climate-change projections. To illustrate the first point, we focus on the large-scale, wind-driven flow of the mid-latitude oceans which contribute in a crucial way to Earth's climate, and to changes therein. We study the low-frequency variability (LFV) of the wind-driven, double-gyre circulation in mid-latitude ocean basins, via the bifurcation sequence that leads from steady states through periodic solutions and on to the chaotic, irregular flows documented in the observations. This sequence involves local, pitchfork and Hopf bifurcations, as well as global, homoclinic ones. The natural climate variability induced by the LFV of the ocean circulation is but one of the causes of uncertainties in climate projections. Another major cause of such uncertainties could reside in the structural instability in the topological sense, of the equations governing climate dynamics, including but not restricted to those of atmospheric and ocean dynamics. We propose a novel approach to understand, and possibly reduce, these uncertainties, based on the concepts and methods of random dynamical systems theory. As a very first step, we study the effect of noise on the topological classes of the Arnol'd family of circle maps, a paradigmatic model of frequency locking as occurring in the nonlinear interactions between the El Nino-Southern Oscillations (ENSO) and the seasonal cycle. It is shown that the maps' fine-grained resonant landscape is smoothed by the noise, thus permitting their coarse-grained classification. This result is consistent with stabilizing effects of stochastic parametrization obtained in modeling of ENSO phenomenon via some general circulation models.Comment: Invited survey paper for Special Issue on The Euler Equations: 250 Years On, in Physica D: Nonlinear phenomen

Nonautonomous dynamical systems: from theory to applications

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 27-04-201

Dimension of Scrambled Sets and The Dynamics of Tridiagonal Competitive-Cooperative System

One of the central problems in dynamical systems and differential equations is the analysis of the structures of invariant sets. The structures of the invariant sets of a dynamical system or differential equation reflect the complexity of the system or the equation. For example, any omega-limit set of a finite dimensional differential equation is a singleton implies that each bounded solution of the equation eventually stabilizes at some equilibrium state. In general, a dynamical system or differential equation can have very complicated invariant sets or so called chaotic sets. It is of great importance to classify those systems whose minimal invariant sets have certain simple structures and to characterize the complexity of chaotic type sets in general dynamical systems. In this thesis, we focus on the following two important problems: estimates for the dimension of chaotic sets and stable sets in a finite positive entropy system, and characterizations of minimal sets of nonautonomous tridiagonal competitive-cooperative systems