A tutela dos interesses metaindividuais em juízo e a busca de mecanismos que assegurem que o processo seja instrumento para o acesso substancial à justiça inserem-se no contexto das ondas renovatórias do processo civil. Embora o movimento em questão situe-se cronologicamente na década de 1960, o ordenamento jurídico pátrio ainda não fez fluir simultaneamente as três ondas. Os denominados interesses metaindividuais abrangem tanto os interesses essencialmente coletivos, nos quais se incluem os difusos e coletivos, quanto os interesses doutrinariamente denominados de acidentalmente coletivos, correspondentes, nos termos do microssistema processual coletivo, ao conceito de interesse individual homogêneo. No Brasil, a tutela jurisdicional de interesses metaindividuais pode se dar através do processo coletivo e, também, por meio do processo civil tradicional, caso apresentem viés estritamente individual. A dificuldade de enquadramento de determinados interesses dentro dos conceitos legais dados pelo microssistema processual coletivo é recorrente no cotidiano jurisprudencial. O processo civil vigente não apresenta mecanismo normativo apto a evitar a concomitância entre ações individuais e coletivas sobre uma mesma questão jurídica, razão pela qual o sistema convive com demandas de massa repetitivamente levadas ao Poder Judiciário. O processo coletivo, no entanto, não é a única alternativa processual existente para o manejo racional de processos repetitivos: as denominadas ações de grupo prestam-se à resolução isonômica de tais demandas sem que, para tanto, sejam necessárias ficções representativas e grandes debates acerca da legitimidade. O presente trabalho presta-se à análise de tais possibilidades, tendo por norte a otimização do alcance subjetivo da decisão judicial que trate de interesse metaindividual. Encontra-se em trâmite na Câmara dos Deputados o projeto do Novo Código de Processo Civil, que traz, dentre as inovações destinadas ao tratamento de demandas de massa, o incidente de resolução de demandas repetitivas, a centralização de processos repetitivos e a conversão da ação individual em coletiva, mecanismos estes que integrarão o objeto do presente estudo. / The judicial protection of group rights and the search for mechanisms that ensure that civil procedure is actually an instrument for substantial access to justice fall into the context of the waves of reform. Although such motion is chronologically situated in the 1960s, Brazilian Law has not yet fulfilled the task of simultaneously flowing the three waves. The so called group rights refer, in Brazilian Law, both essencially and occasionally group rights. The latter refers, in Brazilian Civil Procedure, to the concept of homogeneous individual rights. In Brazil, the judicial protection of group rights can be led through class actions or individual suits, as long as the right involved has a strictly individual aspect. The difficulty of framing some rights into the legal concepts is iterant in Brazilian Courts daily routine. The current Civil Procedure does not have tools able to avoid the concomitance between individual suits and class actions that refer to the same legal issue. For that reason, the judiciary deals with repetitive suits. Class actions, however, are not the only procedural alternative to rationally dealing with repetitive suits: group actions also aim to adequately resolving suits, but without involving issues as adequacy of representation or further concerns related to standing to sue. This research is dedicated to the analysis of such possibilities, having in mind the optimization of the subjective range of judicial decisions that refer to group rights. The Project of the new Brazilian Civil Procedure Code, currently at the Brazilian Chamber of Deputies, brings, among the innovations that intend to deal with repetitive suits, the incident of resolution of repetitive suits, the centralization of repetitive suits and the possibility of convertion of individual suits into class actions. All of these innovations will also be analised.

This thesis is an account of our study of two branches of dynamical systems theory, namely the mean and pointwise ergodic theory. In our work on mean ergodic theorems, we investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces which includes Fr echet spaces. We show how these theorems simplify the derivation of Mean Ergodic theorems. Next we turn to the topic of pointwise ergodic theorems. We analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of -compact locally compact Hausdor groups acting measure-preservingly on - nite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on di erent Banach function spaces, and how the properties of these function spaces in- uence the weak type inequalities that can be obtained. Finally, we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages. Our investigations of these two parts of ergodic theory are uni ed by the techniques used - locally convex vector spaces, harmonic analysis, measure theory - and by the strong interaction of the nal results, which are obtained in greater generality than hitherto achieved. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2014

This thesis is an account of our study of two branches of dynamical systems theory, namely the mean and pointwise ergodic theory. In our work on mean ergodic theorems, we investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces which includes Fr echet spaces. We show how these theorems simplify the derivation of Mean Ergodic theorems. Next we turn to the topic of pointwise ergodic theorems. We analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of -compact locally compact Hausdor groups acting measure-preservingly on - nite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on di erent Banach function spaces, and how the properties of these function spaces in- uence the weak type inequalities that can be obtained. Finally, we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages. Our investigations of these two parts of ergodic theory are uni ed by the techniques used - locally convex vector spaces, harmonic analysis, measure theory - and by the strong interaction of the nal results, which are obtained in greater generality than hitherto achieved. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2014

Ce mémoire porte sur quelques notions appropriées d'actions de groupe sur les variétés symplectiques, à savoir en ordre décroissant de généralité : les actions symplectiques, les actions faiblement hamiltoniennes et les actions hamiltoniennes. Une connaissance des actions de groupes et de la géométrie symplectique étant prérequise, deux chapitres sont consacrés à des présentations élémentaires de ces sujets. Le cas des actions hamiltoniennes est étudié en détail au quatrième chapitre : l'importante application moment y est définie et plusieurs résultats concernant les orbites de la représentation coadjointe, tels que les théorèmes de Kirillov et de Kostant-Souriau, y sont démontrés. Le dernier chapitre se concentre sur les actions hamiltoniennes des tores, l'objectif étant de démontrer le théorème de convexité d'Atiyha-Guillemin-Sternberg. Une discussion d'un théorème de classification de Delzant-Laudenbach est aussi donnée. La présentation se voulant une introduction assez exhaustive à la théorie des actions hamiltoniennes, presque tous les résultats énoncés sont accompagnés de preuves complètes. Divers exemples sont étudiés afin d'aider à bien comprendre les aspects plus subtils qui sont considérés. Plusieurs sujets connexes sont abordés, dont la préquantification géométrique et la réduction de Marsden-Weinstein. / This Master thesis is concerned with some natural notions of group actions on symplectic manifolds, which are in decreasing order of generality : symplectic actions, weakly hamiltonian actions and hamiltonian actions. A knowledge of group actions and of symplectic geometry is a prerequisite ; two chapters are devoted to a coverage of the basics of these subjects. The case of hamiltonian actions is studied in detail in the fourth chapter : the important moment map is introduced and several results on the orbits of the coadjoint representation are proved, such as Kirillov's and Kostant-Souriau's theorems. The last chapter concentrates on hamiltonian actions by tori, the main result being a proof of Atiyah-Guillemin-Sternberg's convexity theorem. A classification theorem by Delzant and Laudenbach is also discussed. The presentation is intended to be a rather exhaustive introduction to the theory of hamiltonian actions, with complete proofs to almost all the results. Many examples help for a better understanding of the most tricky concepts. Several connected topics are mentioned, for instance geometric prequantization and Marsden-Weinstein reduction.

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L'objectif de ce travail est de décrire des modélisations des points de vue et des changements de points de vue d'images d'un objet planaire dans les algèbres de Clifford conformes. Nous généralisons le modèle conforme de l'espace euclidien à travers une famille à deux paramètres d'horosphère, chacune d'entre elles étant plongée dans un espace vectoriel réel de dimension 4 muni d'une métrique équivalente à la métrique de Minkowski. Nous décrivons par la suite deux approches pour mettre en œuvre ces modèles conformes généralisés pour les représentations d'images. L'idée de base est d'encoder les distorsions perspectives de l'objet causées par la variation du paramètre de latitude de la caméra au travers des paramètres d'une horosphère. La première approche consiste à considérer les horosphères de l'espace de Minkowski de dimension 4 pour encoder les points de vue. Les changements de points de vue sont alors linéarisés à travers un groupe de transformations linéaires et conformes de cet espace. Cette approche est ensuite généralisée en décrivant les points de vue à travers les objets d'un groupoïde dont les morphismes sont des diagrammes commutatifs qui représentent les changements de points de vue. Ainsi, une image conforme est décrite par une application définie sur une horosphère à deux paramètres. L'action du groupoïde sur l'ensemble des images conformes nous conduit à associer à tout objet planaire l'orbite de toutes ses images conformes obtenues à partir de tous les points de vue. / Our purpose in this work is to introduce representations of image viewpoints and viewpoint changes of a planar object in conformal Clifford algebras. Our important preliminary contribution is a generalization of the conformal model of the Euclidean space through a two-parameter family of horospheres. Each one of these is embedded into a real vector space of dimension 4 equipped with a metric equivalent to the Minkowski metric. We describe two approaches that make use of these generalized conformal models for image representations. These are based on modelings of perspective distortions of the object caused by a variation of the latitude angle of the camera. First, we model the image viewpoints by the horospheres of the Minkowski space of dimension 4. In this setting, the viewpoint changes are linearized through a group of linear conformal transformations of this space. This approach is generalized by describing the viewpoints through the objects of a groupoid whose morphisms are commutative diagrams that model the viewpoint changes. A conformal image is then described as a map defined on a horosphere. The action of the groupoid on the set of conformal images leads us to associate with every planar object the orbit of its conformal images from all viewpoints.

Les évolutions sociétales témoignent d’une prise en compte grandissante du concept de victime par le droit, et spécifiquement par le droit de la santé. Le droit de la santé est une branche du droit particulièrement riche et complexe dont l’expansion résulte à la fois d’une judiciarisation des professions de santé et d’une succession de scandales sanitaires. La transversalité du droit de la santé et l’appréhension évidente de la victime conduisent à souligner l’existence d’une relation particulière entre ces deux concepts. Néanmoins, il convient de se demander si le caractère épars des règles du droit de la santé permet d’accorder un statut juridique cohérent à la victime dans ce domaine. Si la recherche d’un statut juridique de la victime peut être entreprise, cela se justifie au regard de la spécificité que le droit de la santé semble lui accorder. La présente étude entreprend de démontrer l’existence d’une singularité de la victime dans le domaine de la santé, légitimant dans le même temps la recherche d’un statut juridique. L’assise de la spécificité de la victime réside dans l’attribution de cette qualité indépendamment de la réalisation d’un risque. La survenance d’un évènement constitue donc un critère temporel dont dépend la qualité de « victime avérée » ou celle de « victime potentielle ». La tangibilité du statut juridique de la victime en droit de la santé naît alors de la reconnaissance de sa singularité à travers la présente dichotomie / Evolutions of the society testify a growing awareness of the concept of victims by the law, and especially in Health Law. Health Law is a branch of law which is particularly rich and complex, and whose expansion results both from the judicialization of the health professions and from the succession of health scandals. The transversality of Health Law and its obvious apprehension of the victim lead to highlight the existence of a special relationship between these two concepts. However, the question is whether the scattered character of the Health Law rules assigns a coherent legal status to the victim in this area. If the search for the legal status of the victim can be undertaken, this can be justified by the specific nature of Health Law. The present study tries to demonstrate the existence of the singularity of the victim in the Health Law, while legitimating at the same time the search for the legal status. The foundation of the victim’s specificity lies in the attribution of this quality independently from the realization of a risk. The occurrence of an event constitutes therefore a temporal criterion on which depends the quality of “proved victim” or that of “potential victim”. The tangibility of the victims’ legal status in Health Law arises from the recognition of the singularity through the present dichotomy

De multiples sanctions répondent à la violation du droit de la consommation. Ces sanctions sont perçues comme des outils indispensables au respect de ce droit. Leur prolifération ainsi que leur caractère dérogatoire au droit commun sont pourtant fréquemment dénoncés. Surpénalisation, prime à la mauvaise foi des consommateurs, automaticité, intrusion du juge dans la sphère contractuelle, atteindraient leur légitimité. L’instrumentalisation des sanctions provoquerait leur confusion et leur dénaturation généralisée. La pénalisation des sanctions civiles et la banalisation des sanctions pénales sont particulièrement critiquées. Mais ce double mouvement est-il systématique ? Ce travail entend apporter une réponse plus nuancée. L’abstraction de la protection offerte aux consommateurs exige certains aménagements. Saisir l’essence des sanctions devient alors nécessaire afin identifier celles qui participent d’un dévoiement. Certaines sanctions seront ainsi réhabilitées. D’autres, à l’inverse, seront condamnées. Après avoir mesuré le dévoiement des sanctions du droit de la consommation, une remise en ordre s’est avérée indispensable. Diverses propositions ont ainsi été formulées, dans une quête constante d’équilibre entre efficacité et validité, afin que l’effectivité du droit de la consommation soit elle-même mieux assurée. / Multiple sanctions respond to violations of consumer law. These sanctions are perceived as essential tools for the respect of this right. However, their proliferation and their derogatory nature from ordinary law are frequently denounced. Over-criminalisation, primacy over bad faith on the part of consumers, automaticity, intrusion by the judge into the contractual sphere, would achieve their legitimacy. The instrumentalization of sanctions would lead to their confusion and general distortion. The criminalisation of civil sanctions and the trivialisation of criminal sanctions are particularly criticised. But is this double movement systematic ? This work aims to provide a more nuanced response. Abstracting from the protection offered to consumers requires certain adjustments. Understanding the essence of sanctions then becomes necessary in order to identify those that participate in a misuse. Some sanctions will thus be rehabilitated. Others, on the other hand, will be condemned. After having measured the misuse of consumer law sanctions, it was essential to put them in order. Many proposals were then formulated, so that the effectiveness of consumer law could be better ensured.