Combinatorial number theory, recurrence of operators and linear dynamics
- Autores: Joan Antoni López Martínez
- Directores de la Tesis: Alfredo Peris Manguillot (dir. tes.), Sophie Grivaux (dir. tes.)
- Lectura: En la Universitat Politècnica de València ( España ) en 2023
- Idioma: inglés
- Tribunal Calificador de la Tesis: José Antonio Bonet Solves (presid.), Luis Bernal González (secret.), Frédéric Bayart (voc.)
- Programa de doctorado: Programa de Doctorado en Matemáticas por la Universitat de València (Estudi General) y la Universitat Politècnica de València
- Materias:
- Enlaces
- Tesis en acceso abierto en: RiuNet
- Resumen
The thesis "Combinatorial Number Theory, Recurrence of Operators and Linear Dynamics" is part of the study of the dynamics of linear operators, simply called Linear Dynamics. The objective of this work is to study multiple notions of recurrence, that linear dynamical systems can present, and which will be classified through Combinatorial Number Theory.
Linear Dynamics studies the orbits generated by the iterations of a linear transformation. The two most studied properties in this branch of mathematics during the last 30 years have been hypercyclicity (existence of dense orbits) and chaos (with its multiple definitions), being this a very active research area with a considerable number of exceptionally deep but also interesting results. We will focus on recurrence, a property widely studied in the classical setting of non-linear dynamical systems, but practically new with respect to Linear Dynamics since it was not until 2014, with the article by Costakis, Manoussos and Parissis entitled "Recurrent linear operators", when this notion started to be systematically studied in the context of operators acting on Banach spaces.
The basic situation from which our study starts is the following: "T : X ---> X" will be a continuous linear operator acting on an F-space "X", although sometimes we will need the underlying space X to be a Fréchet, Banach or Hilbert space. Given a vector "x" and a neighbourhood "U" of "x" we will study the return set "N_T(x,U) = { n : T^n(x) is in U }" and depending on its size, observed from the Combinatorial Number Theory point of view, we will say that the vector "x" presents one property of recurrence or another.
The thesis memoir is a compendium of articles and it has four chapters and one appendix:
1. Adaptation of the revised "author version" of article "Frequently recurrent operators. Journal of Functional Analysis, 283 (12) (2022), paper no. 109713, 36 pages". Here, the strong notions of reiterative, U-frequent and frequent recurrence are defined for the first time, and their basic properties are studied. The theory is finally generalized through the concept of F-recurrence, which is connected to the notion of F-hypercyclicity.
2. Adaptation of the revised "author version" of article "Recurrence properties: An approach via invariant measures. Journal de Mathématiques Pures et Appliquées, 169 (2023), 155-188". In this chapter the recurrence properties for linear operators are related to Ergodic Theory and measure preserving systems.
3. Adaptation of the revised "author version" of the preprint "Questions in linear recurrence: From the T+T-problem to lineability". We solve in the negative an open problem posed in 2014: Let "T : X ---> X" be a recurrent operator. Is it true that the operator "T+T" is recurrent on "X+X"? In order to do that we establish the analogous notion, for recurrence, to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and then we construct recurrent but not quasi-rigid operators on every separable infinite-dimensional Banach space.
4. Adaptation of the revised "author version" of the preprint "Recurrent subspaces in Banach spaces". In this chapter we study the spaceability (existence of an infinite-dimensional closed subspace) for the set of recurrent vectors.
- Appendix. Looking for a self-contained text we have added an appendix with some of the basic Combinatorial Number Theory results that are taken for granted along the different chapters/articles forming this memoir.
Following the regulations established by the Doctoral School the next sections are also included:
- Introduction;
- General discussion of the results;
- Conclusions.
