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

No presente trabalho apresentamos dois teoremas obtidos por Gorak em 2011, que são generalizações para o Teorema de Banach-Stone, envolvendo uma classe de funções não-necessariamente lineares, denominadas quasi-isometrias. / In this work we present two theorems proved by Gorak in 2011. These results are generalizations of the Banach-Stone Theorem envolving a class of not-necessarily linear functions, called quasi-isometries.

No presente trabalho apresentamos dois teoremas obtidos por Gorak em 2011, que são generalizações para o Teorema de Banach-Stone, envolvendo uma classe de funções não-necessariamente lineares, denominadas quasi-isometrias. / In this work we present two theorems proved by Gorak in 2011. These results are generalizations of the Banach-Stone Theorem envolving a class of not-necessarily linear functions, called quasi-isometries.

Cette thèse est consacrée à la théorie spectrale de quelques opérateurs de transport et le comportement asymptotique (pour les temps grands) des solutions des problèmes de Cauchy gouvernés par ces derniers. Dans la première partie, on s'est intéressé aux propriétés spectrales des opérateurs d'advection et de transport des neutrons dans le cadre multidimensionnel pour des conditions aux limites générales. Après avoir établi un résultat de compacité de type lemmes de moyenne indispensable dans notre analyse, on a donné entre autre une description fine du spectre asymptotique de l'opérateur de transport. Ce travail a été complété par l'étude des propriétés de régularité et le comportement asymptotique de la solution du problème de Cauchy gouverné par l'opérateur de transport étudié précédemment pour des conditions aux limites de type bounce-back plus un opérateur compact dans l'espace L^1. Ensuite, on a étudié le caractère bien posé et le comportement asymptotique de la solution d'une équation de transport des neutrons avec des sections efficaces non bornées. Contrairement à la première partie, l'analyse de ce problème nécessite l'usage d'une théorie de perturbation de Miyadera-Voigt pour les opérateurs non bornés. La dernière partie de ce travail porte sur un problème linéaire issu d'un modèle introduit en 1974 par Lebowitz et Rubinow décrivant la prolifération d'une population de cellules structuré par l'âge et la longueur du cycle. Notre analyse a porté sur le cas où la longueur du cycle maximale est infinie. / This thesis is devoted to the spectral theory and the time asymptotic behavior of the solution to Cauchy problems governed by various transport operators. In the first part, we discussed the spectral properties of streaming and transport operators in finite bodies with general boundary conditions. After establishing a compactness result essential to our analysis, we gave a fine description of the asymptotic spectrum of the transport operator. We also derive the regularity and the asymptotic behavior of the solution to Cauchy problem governed by the transport operator supplemented by bounce-back boundary conditions plus a compact operator in the space L^1. In the second part, we discussed the well-posedness and the asymptotic behavior of the solution to Cauchy problem governed by a singular transport operator. Unlike the first part, the analysis of this problem requires the use of Miyadera-Voigt perturbation theory for unbounded operators. In the last part of this work, a Cauchy problem governed by a linear operator introduced by Lebowitz and Rubinow describing a proliferating cell population structured by age and the cycle length was considered. Here our analysis was devoted to the case where the maximum cycle length is infinite.

Notre principal but est le probleme de l'unicite pour les operateurs de diffusion dans $L^\infty$. Ce travail commence par un etude des $C_0$-semi-groupes et des semi-groupes integres dans un contexte tres general. Nous etudions les $C_0$-semi-groupes sur un espace localement convexe et nous introduisons une nouvelle topologie sur l'espace dual tel que l'adjoint d'un $C_0$-semi-groupe est de classe $C_0$ par rapport a cette topologie. Les resultats les plus importants sont un theoreme de caracterisation d'un core du generateur et un theoreme de caracterisation complet d'un generateur essentiel sur un espace localement convexe. Finalement, nous presentons quelques exemples des generateurs essentiels dans $L^\infty$. Dans cette these ont ete obtenues pour la premiere fois la $L^\infty$-unicite des operateurs de Schroedinger et des operateurs de Schroedinger generalises sur une variete riemannienne complete, ainsi que $L^1$-unicite des solutions faibles pour l'equation de transport de masse.