Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2016-06-02T21:00:54Z No. of bitstreams: 2 Dissertação - Jeidy Johana Jimenez Ruiz - 2016.pdf: 946402 bytes, checksum: 0a36384eddfdcc5620d74725a24dd86a (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-06-03T11:43:02Z (GMT) No. of bitstreams: 2 Dissertação - Jeidy Johana Jimenez Ruiz - 2016.pdf: 946402 bytes, checksum: 0a36384eddfdcc5620d74725a24dd86a (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Made available in DSpace on 2016-06-03T11:43:02Z (GMT). No. of bitstreams: 2 Dissertação - Jeidy Johana Jimenez Ruiz - 2016.pdf: 946402 bytes, checksum: 0a36384eddfdcc5620d74725a24dd86a (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) Previous issue date: 2016-03-04 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / The study under existence and uniqueness of limit cycles of equations systems differential is a very active research topic in the qualitative theory of dynamical systems. In this theme we study this topic in discontinuous dynamic systems. Let’s make this in Liénard differentials equation systems, allowing a line of discontinuity. Furthermore, we present the known method of Averaging firstly in your classic version, that is, for class fields at least C2, we study also to generalized version, to piecewise- smooth dynamical systems. As a result, we use this tool to determine the number of limit cycles that can bifurcate of a planar center, inside the equation Liénard differentials equation class. / O estudo sobre existência e unicidade de ciclos limites de sistemas de equações diferenciais é um tópico de grande interesse na teoria qualitativa de sistemas dinâmicos. Nesta dissertação, estudamos este tópico em sistemas dinâmicos descontínuos. Vamos fazer esta análise em sistemas de equações diferenciais de Liénard, permitindo uma linha de descontinuidade. Além disso, vamos apresentar o conhecido método Averaging de primeira ordem, em primeiro lugar na sua versão clássica, isto é, para campos de classe pelo menos C2, depois apresentaremos também a versão generalizada, para sistemas diferenciais definidos por partes. Como resultado, fazemos uso desta ferramenta para determinar o número de ciclos limites que podem bifurcar de um centro planar, dentro da classe de equações diferenciais de Liénard.

Dans ce mémoire, nous étudions l'entropie des systèmes dynamiques différentiables définis sur des variétés riemanniennes non compactes. Dans un premier temps, nous éclaircissons les liens entre différentes notions d'entropie dans ce cadre non compact. Ensuite, nous utilisons ces premiers résultats pour y étudier la validité de l'inégalité de Ruelle. Rappelons ici que cette inégalité, pour des difféomorphismes de variétés riemanniennes compactes, nous dit que l'entropie est majorée par la somme des exposants de Lyapounov positifs. Nous montrons que, lorsque nous enlevons l'hypothèse de compacité, l'inégalité de Ruelle n'est pas toujours satisfaite. Nous obtenons ce résultat en construisant une famille explicite de contre-exemples. En revanche, nous montrons, dans le cas d'un difféomorphisme de comportement asymptotique linéaire, ou du flot géodésique sur le fibré unitaire tangent d'une variété riemannienne à courbure négative, que l'inégalité de Ruelle est toujours satisfaite. Pour finir, nous nous intéressons au problème de la perte possible de masse d'une suite de mesures de probabilité d'une variété riemannienne non compacte. Dans le cas du flot géodésique, nous montrons que l'entropie permet de contrôler la masse d'une limite vague de mesures de probabilité invariantes par le flot pour une classe particulière de variétés géométriquement finies. Plus précisément, nous montrons qu'une suite de mesures d'entropie assez grande ne peut pas perdre la totalité de sa masse. De plus, le minorant optimal de l'entropie dans ce résultat est lié à la géométrie de la partie non compacte de la variété: c'est l'exposant critique maximal des sous-groupes paraboliques du groupe fondamental. / In this work, we study the entropy of smooth dynamical systems defined on non compact Riemannian manifolds. First, we clarify some relations between different notions of entropy in this setting. Second, we use these first results in order to study the validity of Ruelle's inequality. This inequality, for diffeomorphisms defined on compact Riemannian manifolds, says that the measure-theoretic entropy is bounded from above by the sum of the positive Lyapunov exponents. We show that without the compactness assumption, Ruelle's inequality is not always satisfied. We obtain this result by constructing an explicit family of counterexamples. On the other hand, we prove, in the case of diffeomorphisms with linear asymptotic behavior, or that one of the geodesic flow on the unit tangent bundle of a Riemannian manifold with negative curvature, that Ruelle's inequality is always satisfied. Finally, we are interested in the problem of the possible escape of mass of a sequence of probability measures on a non compact Riemannian manifold. In the case of the geodesic flow, we show that the entropy allows to control the mass of a weak$^\ast$-limit of a sequence of probability measures, on the unit tangent bundle of a particular class of geometrically finite manifolds, which are also invariant by the flow. More precisely, we show that a sequence of measures with large enough entropy cannot lose the whole mass. Moreover, the optimal lower bound of the entropy in this result is related to the geometry of the non compact part of the manifold: it is the maximal critical exponent of the parabolic subgroups of the fundamental group.

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In this paper we study local rigidity of actions of simply connected Lie groups. In particular, we apply the Nash-Moser inverse function theorem to give sufficient conditions for the action of a simply connected Lie group to be locally rigid. Let $G$ be a Lie group, $H < G$ a simply connected subgroup and $\Gamma < G$ a cocompact lattice. We apply the result for general actions of simply connected groups to obtain sufficient conditions for the action of $H$ on $\Gamma\backslash G$ by right translations to be locally rigid. We also discuss some possible applications of this sufficient condition / I den här texten så studerar vi lokal rigiditet av gruppverkan av enkelt sammanhängande Liegrupper. Mer specifikt, vi applicerar Nash-Mosers inversa funktionssats för att ge tillräckliga villkor för att en gruppverkan av en enkelt sammanhängande grupp ska vara lokalt rigid. Låt $G$ vara en Lie grupp, $H < G$ en enkelt sammanhängande delgrupp och $\Gamma < G$ ett kokompakt gitter. Vi applicerar resultatet för generella gruppverkan av enkelt sammanhängande grupper för att få tillräckliga villkor för att verkan av $H$ på $\Gamma\backslash G$ med translationer ska vara lokalt rigid. Vi diskuterar också några möjliga tillämpningar av det tillräckliga villkoret.

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