A ideia deste trabalho é apresentar a conjetura de Boyland para o anel e mostrar algums resultados nessa direção. Tal conjectura diz que: Dado um homeomorfismo irrotacional do anel, que possui uma medida com número de rotação positivo, é verdade que, neste caso, existem pontos com número de rotação negativo? Para dar uma resposta parcial a esta pregunta, nesta dissertação (baseada no estudo do [7]) começamos considerando os homeomorfismos do anel que preservam orientação, as componentes de fronteira, com número de rotação positivos em ambas fronteiras, e que tem un levantamento transitivo (o motivo desta hipoteses vem de [3]), mostrando que neste caso 0 está no interior do conjunto de rotação. Este resultado vai permitir provar a conjetura para os homeomorfismos do anel irrotacionais, sem pontos fixos na fronteira e com um levantamento transitivo. Além disso vai permitir estudar a dinâmica de tais homeomorfismos. No final do trabalho, estendemos algums dos teoremas provados ao longo dos capítulos anteriores a um conjunto maior de homeomorfismos e estudamos o comportamento de tais homeomorfismos com base nestes resultados. / The idea of this work is to present Boyland´s Conjecture for the annulus and show some results in its direction. The conjecture is the following: Given a homeomorphism of the annulus, which has a measure with positive rotation number, is it true that, in this case, there are points with negative rotation number?. To give a partial answer to this question, in this dissertation (based on [7]) we begin considering the homeomorphisms of the annulus that preserve orientation and boundary components, with positive rotation numbers in the boundaries, with has a transitive lift (the reason for this hypothesis is in [3]), and we show that 0 is in the interior of the rotation set. This result will be of help to prove the Boyland´s Conjecture for rotationless homeomorphisms of the annulus, without fixed points in the boundaries and with a transitive lift. In addition, we will be able to study the dynamics of such homeomorphisms. In the end of this work, we extend some of the theorems proved in the previous chapters to a bigger set of homeomorphisms and we study the behavior of such homeomorphisms using these results.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Bibliography: leaves 155-156. / by Robert Clack Penner. / Ph.D.

A ideia deste trabalho é apresentar a conjetura de Boyland para o anel e mostrar algums resultados nessa direção. Tal conjectura diz que: Dado um homeomorfismo irrotacional do anel, que possui uma medida com número de rotação positivo, é verdade que, neste caso, existem pontos com número de rotação negativo? Para dar uma resposta parcial a esta pregunta, nesta dissertação (baseada no estudo do [7]) começamos considerando os homeomorfismos do anel que preservam orientação, as componentes de fronteira, com número de rotação positivos em ambas fronteiras, e que tem un levantamento transitivo (o motivo desta hipoteses vem de [3]), mostrando que neste caso 0 está no interior do conjunto de rotação. Este resultado vai permitir provar a conjetura para os homeomorfismos do anel irrotacionais, sem pontos fixos na fronteira e com um levantamento transitivo. Além disso vai permitir estudar a dinâmica de tais homeomorfismos. No final do trabalho, estendemos algums dos teoremas provados ao longo dos capítulos anteriores a um conjunto maior de homeomorfismos e estudamos o comportamento de tais homeomorfismos com base nestes resultados. / The idea of this work is to present Boyland´s Conjecture for the annulus and show some results in its direction. The conjecture is the following: Given a homeomorphism of the annulus, which has a measure with positive rotation number, is it true that, in this case, there are points with negative rotation number?. To give a partial answer to this question, in this dissertation (based on [7]) we begin considering the homeomorphisms of the annulus that preserve orientation and boundary components, with positive rotation numbers in the boundaries, with has a transitive lift (the reason for this hypothesis is in [3]), and we show that 0 is in the interior of the rotation set. This result will be of help to prove the Boyland´s Conjecture for rotationless homeomorphisms of the annulus, without fixed points in the boundaries and with a transitive lift. In addition, we will be able to study the dynamics of such homeomorphisms. In the end of this work, we extend some of the theorems proved in the previous chapters to a bigger set of homeomorphisms and we study the behavior of such homeomorphisms using these results.

Dados dois conjuntos compactos e convexos K1, K2 em R², queremos saber se existem f e h, dois homeomorfismos de T², homotópicos à identidade, que comutam, com levantamentos F e H, tais que K1 e K2 são os seus conjuntos de rotação, respectivamente. Neste trabalho, mostramos alguns casos onde isto não pode acontecer, assumindo restrições nos conjuntos de rotação. Além disso, introduzimos o conceito de conjunto de rotação para semigrupos Abelianos finitamente gerados por homeomorfismos homotópicos à identidade, mostrando um caso em que o semigrupo é anular. / Let K1, K2 in R² be two convex, compact sets. We would like to know if there are commuting homeomorphisms f and h of T², homotopic to the identity, with lifts F and H, such that K1 and K2 are their rotation sets, respectively. In this work, we proof some cases where it cannot happen, assuming some restrictions on rotation sets. Besides that, we introduce the concept of rotation set for Abelian semi-groups finitely generated by homeomorphisms homotopic to the identity, showing a case where the semi-group is annular.

Seja f : T2 -> T2 um homeomorfismo homotópico a identidade e F : R2 -> R2 um levantamento de f tal que seu conjunto de rotação rho(F) é um segmento vertical não degenerado contido em 0 × R. Provamos que se f é ergódico com respeito a medida de Lebesgue no toro e se o vetor de rotação médio (com respeito a mesma medida) é da forma (0, alpha) para alpha em R\\Q então existe M > 0 tal que |(Fn (x) - x)1| <= M para todo x em R2 e n em Z (onde (.)1 :R2 -> R é definida por (x,y)1 =x). / Let f : T2 -> T2 be a homeomorphism homotopic to the identity and F : R2 -> R2 a lift of f such that the rotation set rho(F) is a non-degenerated vertical line segment contained in 0 × R. We prove that if f is ergodic with respect to the Lebesgue measure on the torus and the average rotation vector (with respect to same measure) is of the form (0, alpha) for alpha in R\\Q then there exists M > 0 such that |(Fn (x) - x)1| <= M for all x in R2 and n in Z (where (.)1 :R2 -> R is defined by (x, y)1 = x).

We give conditions which, given two Bernoulli trial measures, determine whether there exists a homeomorphism of Cantor space which sends one measure to the other, answering a question of Oxtoby. We then provide examples, relating these results to the notions of good and refinable measures on Cantor space.

Dans cette thèse nous nous intéressons à la dynamique locale autour d'une sous-variété compacte invariante et à la théorie du nombre de rotation. Dans [Nai82] V. A. Naishul' a montré que parmi les difféomorphismes du plan isotopes à l'identité qui fixent 0, qui préservent l'aire (ou analytiques) et dont la différentielle en $0$ est une rotation, l'angle de cette rotation est un invariant de conjugaison topologique. Ce résultat de Na\u{\i}shul$'$, a été généralisé dans plusieurs directions (voir [GP95], [GLP96] et [Pon12]). Par exemple en dimension supérieure, dans [GP95] J.-M. Gambaudo et E. Pécou ont considéré des difféomorphismes de $\R^{n+2}$ qui possèdent un tore $\T^n$ de dimension $n$ invariant dont la dynamique est topologiquement conjuguée à une rotation irrationnelle. Ils ont défini un nombre de rotation et ont démontré que ce nombre est invariant de conjugaison topologique (par exemple lorsque le difféomorphisme préserve un volume). Dans la première partie du deuxième chapitre de cette thèse, nous proposons d'introduire une notion d'ensemble de rotation local pour les homéomorphismes locaux qui préservent une sous-variété compacte de codimension $2$ dont le fibré normal est trivial. A l'aide de cet ensemble, nous déduirons un résultat qui généralise les travaux en dimension supérieure cités plus haut. Dans [Rue85] D. Ruelle a considéré des difféomorphismes d'une surface dont le fibré tangent est trivial qui préservent une mesure. Il leur a associé un nombre réel qui a été appelé l'invariant de Ruelle. Les constructions de cette thèse nous permettront de voir cet invariant comme un ensemble de rotation local au-dessus d'une mesure. A l'aide de l'invariance par conjugaison de cet ensemble de rotation, nous allons retrouver, à la fin du deuxième chapitre, le résultat démontré par J.-M. Gambaudo et E. Ghys dans [GG97] : l'invariant de Ruelle est en fait invariant de conjugaison topologique. Soit $Homeo_0(\R^2;0)$ l'ensemble des homéomorphismes du plan $\R^2$ isotopes a l'identité qui fixent l'origine $0\in\R^2$. Récemment dans [LeR13], F. Le Roux a donné une définition de l'ensemble de rotation local autour de $0$ d'une isotopie dans $Homeo_0(\R^2;0)$ issue de l'identité, et il a posé la question suivante : cet ensemble est-il toujours un intervalle ? Dans le troisième chapitre de cette thèse, nous allons donner une réponse positive à cette question et aussi à la question analogue dans le cas de l'anneau ouvert. / In this thesis we are interested in the local dynamics around of a compact invariant sub-manifold and in the rotation number theory. In [Nai82] V.A Naihul' proved that, among analytic or area preserving diffeomorphisms in the plane which are isotopic to the identity fix $0$ and whose derivative at $0$ is a rotation, the angle of this rotation is invariant by topological conjugation. This result of Naishul' was generalized in many directions (see [GP95], [GLP96] and [Pon12]). For example in [GP95] J.-M. Gambaudo and E. Pécou considered diffeomorphisms in $\R^{n+2}$, which possess an invariant $n$-dimensional torus $\T^n$ whose dynamics restricted to the torus is topologically conjugate to an irrational rotation. They defined a rotation number, and proved that this number is invariant by topological conjugation among volume-preserving maps. In the first part of the second chapter of this thesis, we propose to introduce a notion of local rotation set for local homeomorphisms, which preserve a compact sub-manifold of codimension 2 whose normal bundle is trivial. Using this set, we will deduce a result which generalizes the above mentioned works. In [Rue85] D. Ruelle considered measure preserving diffeomorphisms of a surface whose tangent bundle is trivial. He associated to them a real number called the Ruelle invariant. The constructions made in this thesis will permit us to see this number as a local rotation set over a measure. The invariance by topological conjugation of this set will us permit, at the end of the second chapter, to prove the following result due to J.-M- Gambaudo and E. Ghys: the Ruelle invariant is invariant by topological conjugacy. Let $Homeo_0(\R^2;0)$ be the set of all homeomorphisms of the plane isotopic to the identity and which fix $0$. Recently in [LeR13] F. Le Roux gave the definition of the local rotation set around of 0 of a general isotopy $I$ in $Homeo_0(\R^2;0)$ from the identity to a homeomorphism $f$ and he asked if this set is always an interval. In the third chapter of this thesis we give a positive answers to this question and to the analogous question in the case of the open annulus.

This dissertation explores geometric measure theory; the first part explores a question posed by Paul Erdös -- Is there a number c > 0 such that if E is a Lebesgue measurable subset of the plane with λ²(E) (planar measure)> c, then E contains the vertices of a triangle with area equal to one? -- other related geometric questions that arise from the topic. In the second part, "we parametrize the theorems from general topology characterizing the continuous images and the homeomorphic images of the Cantor set, C" (abstract, para. 5).

Neste trabalho estudamos o valor mínimo da entropia topológica para uma classe de aplicações isotópicas à identidade em superfícies orientáveis (sem bordo, não necessariamente compactas e possivelmente de tipo finito) sob um ponto de vista estritamente topológico. Este estudo é feito utilizando a nova teoria de forcing para trajetórias transversas de Le Calvez e Tal que se baseia na teoria de Brouwer equivariante, em que é possível folhear superfícies com folhas relacionadas a teoria de Brouwer no plano. O principal resultado deste trabalho é uma melhora na estimativa da entropia topológica obtida por Le Calvez e Tal em um recente trabalho em que os autores buscam ferraduras topológicas em superfícies orientáveis utilizando ferramentas similares apresentadas aqui. Uma aplicação deste resultado acima é feita utilizando aplicações em S^2 que possuam um ponto fixo cuja trajetória pela isotopia deste ponto não seja homotópica a um múltiplo de um loop simples. Com estas hipóteses, melhoramos a estimativa dada por Le Calvez e Tal em que é encontrado um valor mínimo estritamente positivo para a entropia topológica desta aplicação. / In this work we study the minimum topological entropy value for one class of maps isotopics to the identity in oriented surfaces (without border, not necessary compacts and possibly of finite type) under the point of view strictly topological. This study is done using the new forcing theory to transverse trajectories from Le Calvez and Tal which it is based to equivariant Brouwer Theory, on what it is possible to leaf surfaces with leaves related to plane Brouwer theory. The main result in this work is a improvement in the estimates from the topological entropy obtained by Le Calvez and Tal in one recent work where the authors seek topological horseshoes on oriented surfaces using tools very similar to that are shown here. One application of the above result is done using maps on S^2 that have a fixed point whose trajectory by the isotopy of this point do not be homotopic to a multiple of a simple loop. With these hypotheses, we improve the estimates given by Le Calvez and Tal on what is found a strictly positive minimum value to the topological entropy of this map.

viii, 113 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / This dissertation consists of four principal parts. In the first, we introduce the tracial quasi-Rokhlin property for an automorphism α of a C *-algebra A (which is not assumed to be simple or to contain any projections). We then prove that under suitable assumptions on the algebra A , the associated crossed product C *-algebra C *([Special characters omitted.] , A , α) is simple, and the restriction map between the tracial states of C *([Special characters omitted.] , A , α) and the α-invariant tracial states on A is bijective. In the second part, we introduce a comparison property for minimal dynamical systems (the dynamic comparison property) and demonstrate sufficient conditions on the dynamical system which ensure that it holds. The third part ties these concepts together by demonstrating that given a minimal dynamical system ( X, h ) and a suitable simple C *-algebra A , a large class of automorphisms β of the algebra C ( X, A ) have the tracial quasi-Rokhlin property, with the dynamic comparison property playing a key role. Finally, we study the structure of the crossed product C *-algebra B = C *([Special characters omitted.] , C ( X , A ), β) by introducing a subalgebra B { y } of B , which is shown to be large in a sense that allows properties B { y } of to pass to B . Several conjectures about the deeper structural properties of B { y } and B are stated and discussed. / Committee in charge: Christopher Phillips, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Huaxin Lin, Member, Mathematics; Marcin Bownik, Member, Mathematics; Van Kolpin, Outside Member, Economics