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Doctoral Thesis
DOI
10.11606/T.45.2016.tde-24082016-115504
Document
Author
Full name
Rodrigo Manoel Dias Andrade
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2015
Supervisor
Committee
Proença, Rodrigo Bissacot (President)
Abrahamian, Roberto Markarian
Carvalho, Sônia Pinto de
Freire Junior, Ricardo dos Santos
Tal, Fabio Armando
Title in Portuguese
Propriedade de Bernoulli para bilhares hiperbólicos com fronteiras focalizadoras quase planas
Keywords in Portuguese
Bilhares hiperbólicos
Ergodicidade
Propriedade de Bernoulli
Abstract in Portuguese
Neste trabalho, mostramos que os bilhares hiperbólicos construídos originalmente por Bussolari- Lenci têm a propriedade de Bernoulli. Tais bilhares não satisfazem as técnicas standard de Wojtkowski-Markarian-Donnay-Bunimovich para bilhares focalizadores hiperbólicos, a qual requer que o diâmetro da mesa do bilhar seja de mesma ordem que o maior raio de curvatura ao longo da componente focalizadora. Nossa prova, utiliza um teorema ergódico local que nos diz que sob certas condições, existe um conjunto de medida total do espaço de fase do bilhar tal que cada ponto desse conjunto possui uma vizinhança contida (mod 0) em uma componente Bernoulli da aplicação do bilhar.
Title in English
Bernoulli property for hyperbolic billiards with nearly flat focusing boundaries.
Keywords in English
Bernoulli property
Ergodicity
Hyperbolic billiards
Abstract in English
In this work, we show that hyperbolic billiards constructed originally by Bussolari-Lenci has the Bernoulli property. These billiards do not satisfy the standard Wojtkowski-Markarian-Donnay- Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane, which requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. Our proof employs a locally ergodic theorem which says that under a few conditions, there exists a full measure set of the billiard phase space such that each of its points has a neighborhood contained, up to a zero measure set, in one Bernoulli component of the billiard map.
 
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TESE_Rodrigo_Andrade.pdf (878.48 Kbytes)
Publishing Date
2016-09-09
 
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