We investigate the genericity of measure-preserving actions of the free group Fn, on possibly countably infinitely many generators, acting on a standard probability space. Specifically, we endow the space of all measure-preserving actions of Fn acting on a standard probability space with the weak topology and explore what properties may be verified on a comeager set in this topology. In this setting we show an analog of the classical Rokhlin Lemma. From this result we conclude that every action of Fn may be approximated by actions which factor through a finite group. Using this finite approximation we show the actions of Fn, which are rigid and hence fail to be mixing, are generic. Combined with a recent result of Kerr and Li, we obtain that a generic action of Fn is weak mixing but not mixing. We also show a generic action of Fn has sigma-entropy at most zero. With some additional work, we show the finite approximation result may be used to that show for any action of Fn, the crossed product embeds into the tracial ultraproduct of the hyperfinite II1 factor. We conclude by showing the finite approximation result may be transferred to a subspace of the space of all topological actions of Fn on the Cantor set. Within this class, we show the set of actions with sigma-entropy at most zero is generic.

Submitted by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-02-20T14:12:48Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese - Marcos A. Alcantara.pdf: 8400753 bytes, checksum: 2ef1ddefa929adecd87c6c952124ec8b (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-02-20T14:13:03Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese - Marcos A. Alcantara.pdf: 8400753 bytes, checksum: 2ef1ddefa929adecd87c6c952124ec8b (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-02-20T14:13:19Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese - Marcos A. Alcantara.pdf: 8400753 bytes, checksum: 2ef1ddefa929adecd87c6c952124ec8b (MD5) / Made available in DSpace on 2017-02-20T14:13:19Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Tese - Marcos A. Alcantara.pdf: 8400753 bytes, checksum: 2ef1ddefa929adecd87c6c952124ec8b (MD5) Previous issue date: 2015-12-15 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Let M be a compact orientable Riemannian manifold with boundary. We first study in this thesis the transversality of a family of metrics via a family of Bilaplacian opera-tors parametrised by the metrics themselves. We next consider a family of Bilaplacian operators with variation of their dominion in flat Riemannian manifolds and establish a condition of transversality in this case. In the case of arbitrary Riemannian manifold, we obtain that the spectrum of 2g. is simple and the set of parameters is residual. We complete this thesis by studying the generic propeties of the eigenvalues of the Laplacian of a family of hyper-surfaces of revolution in the vector space Rn+1 / Nesta tese primeiramente tratamos de transversalidade de famílias de métricas, em que foi tomada uma família de operadores Bilaplacianos parametrizada pelas métricas de uma variedade Riemanniana orientável M compacta com bordo. Em seguida foi considerada uma família de operadores Bilaplacianos em que o parâmetro é o domínio de definição do operador, no caso da variação do domínio em variedades Riemannianas flat foi mostrada a condição de transversalidade para a família de operadores Bilaplacianos parametrizados por tais domínios. Porém no caso de variedades Riemannianas quaisquer, obtemos a simplicidade genérica dos autovalores associados ao 2g. Por último, estudamos a situação genérica dos autovalores do Laplaciano numa família de hipersuperficies de rotação no espaço vetorial Rn+1.

J’examine les relations entre propriétés des failles géologiques long-termes et propriétés des forts séismes que produisent ces failles. J’ai compilé les données sismologiques disponibles sur les grands séismes historiques mondiaux, et cartographié, sur images satellitaires, les failles long-termes rompues par ces séismes et les traces des ruptures. L’analyse combinée des données montre que : i) les failles long-termes ont certaines propriétés génériques (organisation des réseaux, segmentation latérale, forme de distribution du glissement cumulé, etc) ; ii) les forts séismes ont également des propriétés communes (similarité de distribution du glissement cosismique, du nombre de segments rompus, de la chute de contrainte sur chaque segment majeur rompu, de la distance relative entre hypocentre et zone de glissement maximum, etc) ; iii) la maturité structurale des failles est la propriété tectonique qui impacte le plus le comportement des forts séismes. Il est probable que cette maturité diminue la friction statique et la complexité géométrique du plan de faille. Elle agit sur la localisation de la zone d’initiation du séisme, sur la localisation et l’amplitude maximum du glissement cosismique, sur la direction de décroissance de ce glissement, sur la « capacité » de la rupture à se propager et donc sur sa vitesse de propagation. Elle dicte le nombre de segments majeurs qui peuvent être rompus, et par conséquent, elle contrôle la longueur totale et la chute de contrainte globale de la rupture. Pour comprendre la physique des forts séismes, il apparaît donc indispensable d’analyser conjointement les propriétés des failles rompues et les propriétés des séismes produits. / I examine the relations between the properties of long-term geological faults and the properties of the large earthquakes these faults produce. I have gathered available seismological information on large historical earthquakes worldwide and mapped in detail, on satellite images, both the long-term fault and the rupture traces. The combined analysis of the data shows that: i) long-term faults have a number of generic properties (arrangement of overall fault networks, lateral segmentation of fault traces, form of cumulative slip distribution, etc); ii) large earthquakes also have generic properties (similarity of envelope shape of coseismic slip-length profiles, of decrease in rupture width along rupture length, of number of broken segments, of stress drop on broken segments, of relative distance between hypocenter and zone of maximum slip, etc); iii) the structural maturity of the faults is the tectonic property most impacting the behavior of large earthquakes. The maturity likely acts in reducing both the static friction and the geometric complexity of the fault plane. It partly governs the location of the earthquake initiation, the location and amplitude of the maximum coseismic slip, the direction of the coseismic slip decrease, the rupture propagation efficiency and speed, the number of major fault segments that are broken, and hence the rupture length and its overall stress drop. To understand the physics of earthquakes, it thus seems necessary to analyze jointly the tectonic properties of the broken faults and the seismological properties of the earthquakes.

Let M be a possibly non compact smooth manifold. We study genericity in the C^k topology (3<=k<=+infty) of nondegeneracy properties of semi-Riemannian geodesic flows on M. Namely, we prove a new version of the Bumpy Metric Theorem for a such M and also genericity of metrics that do not possess any degenerate geodesics satisfying suitable endpoints conditions. This extends results of Biliotti, Javaloyes and Piccione for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P of MxM that satisfies an admissibility condition. Immediate consequences are generic non conjugacy between two points and non focality between a point and a submanifold (or also between two submanifolds). / Seja M uma variedade suave possivelmente não compacta. Estuda-se a genericidade na topologia C^k (3<=k<=+infty) de propriedades de não degenerescência de fluxos geodésicos semi-Riemannianos em M. A saber, provase uma nova versão do Teorema de Métricas Bumpy para uma tal M e também a genericidade de métricas que não possuem geodésicas degeneradas cujos pontos finais satisfazem certas condições. Isso estende resultados anteriores de Biliotti, Javaloyes and Piccione para geodésicas com extremos fixos para o caso onde os extremos variam em uma subvariedade compacta P de M ×M que satisfaz uma condição de admissibilidade. Consequências imediatas são genericidade de não conjugação entre dois pontos e não focalidade entre um ponto e uma subvariedade (ou também entre duas subvariedades).

Let M be a possibly non compact smooth manifold. We study genericity in the C^k topology (3<=k<=+infty) of nondegeneracy properties of semi-Riemannian geodesic flows on M. Namely, we prove a new version of the Bumpy Metric Theorem for a such M and also genericity of metrics that do not possess any degenerate geodesics satisfying suitable endpoints conditions. This extends results of Biliotti, Javaloyes and Piccione for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P of MxM that satisfies an admissibility condition. Immediate consequences are generic non conjugacy between two points and non focality between a point and a submanifold (or also between two submanifolds). / Seja M uma variedade suave possivelmente não compacta. Estuda-se a genericidade na topologia C^k (3<=k<=+infty) de propriedades de não degenerescência de fluxos geodésicos semi-Riemannianos em M. A saber, provase uma nova versão do Teorema de Métricas Bumpy para uma tal M e também a genericidade de métricas que não possuem geodésicas degeneradas cujos pontos finais satisfazem certas condições. Isso estende resultados anteriores de Biliotti, Javaloyes and Piccione para geodésicas com extremos fixos para o caso onde os extremos variam em uma subvariedade compacta P de M ×M que satisfaz uma condição de admissibilidade. Consequências imediatas são genericidade de não conjugação entre dois pontos e não focalidade entre um ponto e uma subvariedade (ou também entre duas subvariedades).

Dans cette thèse nous étudions la dynamique engendrée par une famille de flots Hamiltoniens. Un tel système dynamique à plusieurs générateurs est aussi appelé ‘polysystème’. Motivés par des questions liées au phénomène de la diffusion d’Arnold, notre objectif est de construire des trajectoires du polysystème qui relient deux régions lointaines de l’espace des phases. La thèse est divisée en trois parties.Dans la première partie, nous considérons le polysystème engendré par les flots discrétisés d’une famille d’Hamiltoniens Tonelli. En utilisant une approche variationnelle issue de la théorie KAM faible, nous donnons des conditions suffisantes pour l’existence des trajectoires souhaitées.Dans la deuxième partie, nous traitons le cas d’un polysystème engendré par un couple de flots Hamiltoniens à temps continu, dont l’étude rentre dans le cadre de la théorie géométrique du contrôle. Dans ce contexte, nous montrons dans certains cas la transitivité d’un polysystème générique, à l’aide du théorème de transversalité de Thom.La dernière partie de la thèse est dédiée à obtenir une nouvelle version du théorème de transversalité de Thom s’exprimant en termes d’ensembles rectifiables de codimension positive. Dans cette partie il n’est pas question de polysystèmes, ni d’Hamiltoniens. Néanmoins, les résultats obtenus ici sont utilisés dans la deuxième partie de la thèse / In this thesis we study the dynamics generated by a family of Hamiltonian flows. Such a dynamical system with several generators is also called ‘polysystem’.Motivated by some questions related to the phenomenon of Arnold diffusion, our aim is to construct trajectories of the polysystem which connect two far-apart regions of the phase space.The thesis is divided into three parts.In the first part, we consider the polysystem generated by the time-onemaps of a family of Tonelli Hamiltonians. By using a variational approach falling within the framework of weak KAM theory, we give sufficient conditions for the existence of the desired trajectories.In the second part, we address the case of a polysystem generated by twocontinuous-time Hamiltonian flows. This problem fits into the framework of geometriccontrol theory. In this context, we show in some cases the transitivity of a generic polysystem, by means of Thom’s transversality theorem.The third and last part of the thesis is devoted to the proof of a newversion of Thom’s transversality theorem, formulated in terms of rectifiable sets of positive codimension. Neither polysystems nor Hamiltonians are explicitly involved in this part. However, the results obtained here are used in the second part of the thesis.