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- ItemSubshifts on groups and computable analysis(2024) Carrasco Vargas, Nicanor; Rojas González, Luis Cristóbal; Pontificia Universidad Católica de Chile. Facultad de MatemáticasSubshifts are a fundamental class of topological dynamical systems. The study of subshifts on groups different from $\mathbb{Z}$, such as $\mathbb{Z}^d$, $d\geq 2$, has been a subject of intense research in recent years. These investigations have unveiled aremarkable connection between dynamics and recursion theory. That is, different questions about the dynamics of these systems have been answered in recursion-theoretical terms. In this work we further explore this connection. We use the framework of computable analysis to explore the class of effective dynamical systems on metric spaces, and relate these systems to subshifts of finite type (SFTs) on groups. We prove that every effective dynamical system on a general metric space is the topological factor of an effective dynamical system with topological dimension zero. We combine this result with existing simulation results to obtain new examples of systems that are factors of SFTsWe also study a conjugacy invariant for subshifts on groups called Medvedev degree. This invariant is a complexity measure of algorithmic nature. We develop the basic theory of these degrees for subshifts on arbitrary finitely generated groups. Using these tools we are able to classify the values that this invariant attains for SFTs and other classes of subshifts on several groups. Furthermore, we establish a connection between these degrees and the distribution of isolated points in the space of all subshifts. Motivated by the study of Medvedev degrees of subshifts, we also consider translation-like actions of groups on graphs. We prove that every connected, locally finite, and infinite graph admits a translation by $\mathbb{Z}$, and that this action can be chosen transitive exactly when the graph has one or two ends. This generalizes a result of Seward about translation-like action of $\mathbb{Z}$ on finitely generated groups. Our proof is constructive, and allows us to prove that under natural hypotheses, translation-like actions by $\mathbb{Z}$ on groups and graphs can be effectively computed.
- ItemProblems on conformal invariance and Yamabe-Type flows(2024) Espinal Florez, María Fernanda; Sáez Trumper, Mariel; Pontificia Universidad Católica de Chile. Facultad de MatemáticasThis work is specifically focused on the study of quantities in Riemannian geometry under a conformal change of metric, that is, under changes of metric which stretch the length of vectors but preserve the angle between any pair of vectors. In this context, my thesis work has centered on the study of symmetric polynomials σk of the eigenvalues of the Schouten tensor, which satisfy a tranformation law under conformal changes. This work consists of two parts. The rst problem concentrates on Yamabe-type ows for σk-curvature, which are classic examples of intrinsic non-linear geometric ows. Inspired by work of Daskalopoulos and Sesum [22], we investigate the existence and classi cation of conformally at rotationally symmetric k-Yamabe gradient solitons replacing scalar curvature by σk-curvature. Our rst result reduces the classi cation of k-Yamabe solitons to the classi cation of global smooth solutions of a fully nonlinear elliptic equation. Regarding the existence result, through a phase-plane analysis of an autonomous system of ordinary equations as in [71], we were able to prove local existence of the ow under conditions of admissibility for the initial metric when n ≥ 2k. Additionally, we had to analyze the asymptotic behavior and solution pro le in each case, taking into account, especially the admissibility of the solution. In contrast with the classical case, the fully non-linear nature of the problem requires additional restrictions (to ensure admissibility) and a more delicate analysis. On the other hand, in collaboration with Professor M. González [27] we work on the k-Yamabe singular problem. The research was focused on constructing metrics with constant σ2-curvature and non-isolated singularities. Speci cally, we contructed a complete non-compact Riemannian metrics with positive constant σ2-curvature on the sphere Sn with a prescribed singular set Λ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than n−√n−2 2 . This is a fully non-linear problem, nevertheless, we show that the classical gluing method (used by Mazzeo-Pacard for the scalar curvature [56]) still works in this setting since the linearized operator has good mapping properties in weighted spaces. The idea to construct this metric is to nd rst an approximate metric with the right asymptotic behavior near the singularity. Even though many of our arguments would work for a general k, we have some computational di culties that restrict our theorem to k = 2.
- ItemAlmost 1-1 extensions, equicontinuous systems and residually finite groups(2024) Gómez Ortiz, Jaime Andrés; Cortéz, María Isabel; Pontificia Universidad Católica de Chile. Facultad de MatemáticasThe purpose of this document is to present our study on the various properties of almost 1-1 extensions of G-odometers with regards to the realization of Choquet simplices, mean-equicontinuity, and the construction of specific almost 1-1 exten- sions of equicontinuous systems. These systems can be viewed as a topological generalization of equicontinuous systems with diverse behavior on some aspects as entropy, the set of probability invariant measures, and more. Each problem is addressed within a general framework without assuming any amenable property on the acting group, except for the last problem where amenability was essential for constructing a specific type of almost 1-1 extensions. This thesis is divided into three parts, with the first two chapters presenting the results of two different manuscripts that are published and submitted, respectively.
- ItemMétodos DPG para el problema quad-curl(2024) Herrera Ortiz, Pablo; Heuer, Norbert; Führer, Thomas; Pontificia Universidad Católica de Chile. Facultad de MatemáticasLos problemas relevantes de la magnetohidrodinámica y la dispersión electromagnética utilizan operadores diferenciales de cuarto orden de tipo rotacional, generalmente denominados operadores quad-curl. Su uso requiere métodos de aproximación numérica. En el caso de los operadores quad-curl, la literatura correspondiente es escasa. La discretización de operadores de cuarto orden es difícil debido al requisito de regularidad para las aproximaciones conformes y la presencia de kernels no triviales. Proponemos emplear el método de Petrov-Galerkin discontinuo (método DPG) con funciones de test óptimas. Este es un marco propuesto por Demkowicz y Gopalakrishnan que tiene como objetivo la estabilidad discreta automática de los esquemas de aproximación.El trabajo está dividido en tres partes. La primera parte examina el problema quad-div en dos y tres dimensiones, mostrando su relación con el operador quad-curl $Curl^4$ en el caso 2D. Presentamos el problema como sistemas de primer y segundo orden. Adicionalmente, proporcionamos un método completamente discreto y realizamos un experimento numérico para el caso adaptativo. En la segunda parte, escribimos el operador quad-curl como $-\Curl\Delta\Curl$, formulamos el problema como un sistema de segundo orden y proporcionamos una formulación variacional ultra-débil. Utilizamos los operadores de Fortin del método DPG para el problema de Kirchhoff--Love en 2D para analizar el esquema completamente discreto. Mostramos una aplicación al problema de Stokes en 2D con cargas en $L_2$ y $H^{-1}$. En la tercera parte, estudiamos directamente el operador $\Curl^4$ en 3D como un sistema de segundo orden y proporcionamos una formulación variacional ultra-débil. En este caso, la existencia de un operador de Fortin es un problema abierto.A lo largo de la tesis, empleamos el marco teórico DPG con formulaciones ultra-débiles. La mayor parte de nuestro análisis se centra en estudiar los operadores de traza, los espacios de traza y los saltos. Estos son claves para caracterizar la regularidad, la conformidad y las condiciones de contorno. Desarrollamos operadores de Fortin los cuales son necesarios para la estabilidad de las formulaciones mixtas. Para todos los casos definimos y analizamos los operadores de traza y espacios necesarios, demostramos el buen planteamiento de las formulaciones variacionales y su discretización, y derivamos estimaciones de error a priori.También examinamos técnicas para la inclusión de condiciones de contorno no homogéneas.Proporcionamos experimentos numéricos para todos los problemas y formulaciones. Estos confirman las propiedades de convergencia esperadas.
- ItemThe Generalized Torelli Problem through the geometry of the Gauss map(2024) Rahausen Rodríguez, Sebastián Andrés; Auffarth, Robert; Pontificia Universidad Católica de Chile. Facultad de MatemáticasGiven a non-hyperelliptic curve C of genus g and 1