Dado um espaço vetorial topológico $X$ e um operador linear $T$ contínuo em $X$, dizemos que $T$ é {\\it hipercíclico} se, para algum $y \\in X$, o conjunto $\\{y, T(y), T^2(y), T^3(y), \\ldots T^n(y) \\ldots \\}$ for denso em $X$. Um dos principais resultados envolvendo operadores hipercíclicos consiste no chamado {\\it Critério de Hiperciclicidade}. Tal Critério fornece uma condição suficiente para que um operador linear contínuo seja hipercíclico. Por muitos anos, procurou-se saber se o Critério também era uma condição necessária. Em \\cite, Bayart e Matheron construíram, nos espaços de Banach clássicos $c_0$ e $\\ell_p, 1 \\leq p < \\infty$, um operador hipercíclico $T$ que não satisfaz o Critério. Neste trabalho, apresentamos a construção realizada por Bayart e Matheron. Além disso, também apresentamos alguns resultados sobre hiperciclicidade. / Given a topological vector space $X$ and a continuous linear operator $T$, we say that $T$ is {\\it hypercylic} if, for some $y \\in X$, the set $\\{y, T(y), T^2(y), T^3(y), \\ldots T^n(y) \\ldots \\}$ is dense in $X$. One of the main results concerning hypercyclic operators is the so-called {\\it Hypercyclicity Criterion}. Such Criterion gives a sufficient condition to a continuous linear operator be hypercyclic. For many years, it sought to know if the Criterion was also a necessary condition. In \\cite, Bayart and Matheron constructed, in the classical Banach spaces $c_0$ e $\\ell_p, 1 \\leq p < \\infty$, a hypercyclic operator $T$ which doesn\'t satisfy the Criterion. In this work, we present the Bayart/Matheron construction. We also present some results about hypercyclicity.

Dado um espaço vetorial topológico $X$ e um operador linear $T$ contínuo em $X$, dizemos que $T$ é {\\it hipercíclico} se, para algum $y \\in X$, o conjunto $\\{y, T(y), T^2(y), T^3(y), \\ldots T^n(y) \\ldots \\}$ for denso em $X$. Um dos principais resultados envolvendo operadores hipercíclicos consiste no chamado {\\it Critério de Hiperciclicidade}. Tal Critério fornece uma condição suficiente para que um operador linear contínuo seja hipercíclico. Por muitos anos, procurou-se saber se o Critério também era uma condição necessária. Em \\cite, Bayart e Matheron construíram, nos espaços de Banach clássicos $c_0$ e $\\ell_p, 1 \\leq p < \\infty$, um operador hipercíclico $T$ que não satisfaz o Critério. Neste trabalho, apresentamos a construção realizada por Bayart e Matheron. Além disso, também apresentamos alguns resultados sobre hiperciclicidade. / Given a topological vector space $X$ and a continuous linear operator $T$, we say that $T$ is {\\it hypercylic} if, for some $y \\in X$, the set $\\{y, T(y), T^2(y), T^3(y), \\ldots T^n(y) \\ldots \\}$ is dense in $X$. One of the main results concerning hypercyclic operators is the so-called {\\it Hypercyclicity Criterion}. Such Criterion gives a sufficient condition to a continuous linear operator be hypercyclic. For many years, it sought to know if the Criterion was also a necessary condition. In \\cite, Bayart and Matheron constructed, in the classical Banach spaces $c_0$ e $\\ell_p, 1 \\leq p < \\infty$, a hypercyclic operator $T$ which doesn\'t satisfy the Criterion. In this work, we present the Bayart/Matheron construction. We also present some results about hypercyclicity.

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Dado E um espaço vetorial topológico e T um operador linear contínuo em E, diremos que T é hipercíclico se, para algum elemento x pertencente a E, a órbita de x sob T, Orb(x,T)={x, Tx, T^2 x,...}, for densa em E. Nosso objetivo será apresentar alguns resultados sobre hiperciclicidade e observar como alguns espaços comportam-se diante dessa classe de operadores. \\\\ / Let E be a topological vector space and T a continuous linear operator on E. We say that T is hypercyclic if, for some x in E, the orbit of x on T, Orb(x,T)={x, Tx, T^2 x,...}, is dense in E. Our aim will be to study some results about hypercyclicity and to observe how some spaces behave regarding this class of operators.

Dado E um espaço vetorial topológico e T um operador linear contínuo em E, diremos que T é hipercíclico se, para algum elemento x pertencente a E, a órbita de x sob T, Orb(x,T)={x, Tx, T^2 x,...}, for densa em E. Nosso objetivo será apresentar alguns resultados sobre hiperciclicidade e observar como alguns espaços comportam-se diante dessa classe de operadores. \\\\ / Let E be a topological vector space and T a continuous linear operator on E. We say that T is hypercyclic if, for some x in E, the orbit of x on T, Orb(x,T)={x, Tx, T^2 x,...}, is dense in E. Our aim will be to study some results about hypercyclicity and to observe how some spaces behave regarding this class of operators.

Cette thèse est principalement consacrée à des problèmes de dynamique linéaire dans les espaces de Banach. Répondant à une question récente de Hajek et Smith, on construit notamment, dans tout espace de Banach séparable, un opérateur borné tel que ses orbites tendent vers l'infini sur une partie ni vide, ni dense. On relie également, à l'aide d'un autre résultat, le module de lissité asymptotique au comportement des opérateurs bornés. / This work is mainly devoted to some problems of linear dynamics in Banach spaces. In particular, we answer a recent question of Hajek and Smith by constructing, in any separable Banach space, a bounded operator such that its orbits tending to infinity form a set which is neither empty, nor dense. We also connect the behaviour of bounded operators with the asymptotic modulus of smoothness.

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/10251/48519 / TESIS

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Il existe de nombreuses notions de petitesse en analyse. On considère trois d'entre elles: la Haar-négligeabilité, la Gauss-négligeabilité et la sigma-porosité. On étudie à quelles conditions le cône positif d'une base de Schauder est Haar-négligeable, et ce que cela entraîne pour l'espace de Banach associé. On étudie également sous quelles conditions l'ensemble des vecteurs non-hypercycliques d'un opérateur hypercyclique est Haar-négligeable ou sigma-poreux. / There are many notions of smallness in Analysis. We will consider three of them: Haar-negligeability, Gauss-negligeability and sigma-porosity. We will study on which conditions the positive cone of a Schauder basis is Haar-null, and its consequence on the Banach space. We will also study on which conditions the set of non-hypercyclic vectors of an hypercyclic operator is Haar-null or sigma-porous.