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Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2011.tde-18042011-130215
Document
Author
Full name
Bernardo Gabriel Marques
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2011
Supervisor
Committee
Tal, Fabio Armando (President)
Gomes, Bernardo Paschoarelli Veiga
Zanata, Salvador Addas
Title in Portuguese
A conjectura de Boyland para homeomorfismos do anel
Keywords in Portuguese
conjectura de Boyland
conjunto de rotação
Homeomorfismos do anel
transitividade
Abstract in Portuguese
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.
Title in English
Boyland's conjecture for annulus homeomorphisms
Keywords in English
Boyland´s Conjecture.
Homeomorphisms of the Annulus
rotation set
transitivity
Abstract in English
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.
 
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deposito.pdf (733.32 Kbytes)
Publishing Date
2011-05-12
 
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