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Doctoral Thesis
Full name
Uirá Norberto Matos de Almeida
Knowledge Area
Date of Defense
São Carlos, 2018
Apaza, Carlos Alberto Maquera (President)
Barbot, Thierry
Catuogno, Pedro Jose
Tahzibi, Ali
Title in English
Contact Anosov actions with smooth invariant bund
Keywords in English
Algebraic actions
Anosov Actions
Contact Structures
Geometric Structures
Abstract in English
The problem of classifying the Anosov systems is of great interest in the theory of dynamical systems. The most important known examples are of algebraic nature and it has been conjectured on 1960s by S. Smale (SMALE, 1967) that these are in fact the only examples. This conjecture has been proved false for Anosov flows, where counter examples had been constructed for odd dimensional manifolds ((HANDEL; THURSTON, 1980) and (BARTHELMé et al., )). This non algebraic examples however are very pathological, and with some stronger hypothesis, for example, smoothness of the invariant bundles, the conjecture remains open. In 1992, it was published a paper (BENOIST; FOULON; LABOURIE, 1992) which proved that contact Anosov flows with smooth invariant bundles are in fact algebraic. In this monograph we seek to generalize the result obtained in (BENOIST; FOULON; LABOURIE, 1992). For this end, we create an adequate definition for contact Anosov Rk-actions, and following the proof strategy used in (BENOIST; FOULON; LABOURIE, 1992) we obtained a partial generalization of this result.
Title in Portuguese
Ações Anosov de contato com fibrados invariantes suaves
Keywords in Portuguese
Ações algébricas
Ações Anosov
Estruturas de contato
Estruturas geométricas
Abstract in Portuguese
O problema da classificação dos sistemas Anosov são de grande interesse dentro da teoria dos sistemas dinâmicos. Os principais exemplos conhecidos são de natureza algébrica e foi levantada na década de 1960 a conjectura de que estes são os únicos exemplos (SMALE, 1967). Esta conjectura se mostrou falsa para fluxos Anosov (ações de R), onde foram construídos contraexemplos em variedades de dimensões impares ((HANDEL; THURSTON, 1980) e (BARTHELMé et al., )). Estes contra exemplos no entanto são de natureza patológica, e sob hipóteses um pouco mais fortes, por exemplo, suavidade dos fibrados invariantes, a conjectura permanece em aberto. Em 1992, foi publicado um artigo (BENOIST; FOULON; LABOURIE, 1992) provando que fluxos de contato Anosov com fibrados invariantes suaves são de fato algébricos . Neste trabalho procuramos generalizar o resultado obtido em (BENOIST; FOULON; LABOURIE, 1992). Para isso criamos uma definição adequada para ações de Rk contato Anosov, que generalizam a noção de fluxo de contato Anosov, e seguindo a estratégia de prova utilizada em (BENOIST; FOULON; LABOURIE, 1992), obtivemos uma generalização parcial deste resultado.
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