Abundância de operadores lineares transitivos em dimensão infinita.
Data
2022-11-04
Autores
Kawahama, Felipe Hikari 
Orientadores
Cirilo, Patricia Romano
Tipo
Dissertação de mestrado
Título da Revista
ISSN da Revista
Título de Volume
Resumo
A teoria de Sistemas Dinâmicos estuda o comportamento de fenômenos em um conjunto no decorrer do tempo e se interessa majoritariamente em sistemas com um certo grau de desordem e imprevisibilidade. Essas propriedades geralmente não são associadas a ação de operadores lineares em espaços vetoriais e com razão em dimensão finita: não existem operadores transitivos nesses espaços. Contudo, em espaços de dimensão infinita existem operadores com as mesmas propriedades encontradas em dinâmicas contínuas e ergódicas em espaços métricos compactos não lineares. Além dessas propriedades, também é de interesse da área de Sistemas Dinâmicos se propriedades dinâmicas interessantes de funções são mantidas para perturbações suficientemente próximas. Embora seja sabido que no caso linear isso nem sempre seja verdade, é possível definir uma classe aberta de operadores com ricas propriedades dinâmicas quando restringimos seu domínio.
The Dynamical Systems theory studies the behaviour of phenomena acting on sets through time and in most cases it takes interest in systems with a certain degree of unorderedness and unpredictability. These properties are not generally associated to the actions of linear operators in vector spaces and in finite dimensional ones there is a reason for that: there are no transitive operators in these spaces. However, in infinite dimensional vector spaces there are operators with the same dynamical properties as the ones found in continuous and ergodic dynamics in non-linear compact metric spaces. Besides these properties, the Dynamical Systems theory also takes interest in whether interesting dynamical properties of a certain function are preserved under sufficiently small perturbations. Even though it is known that, in the linear context, this is not generally the case, it is possible to define an open class of operators where each one of them has rich dynamical properties when its domain is restricted.
The Dynamical Systems theory studies the behaviour of phenomena acting on sets through time and in most cases it takes interest in systems with a certain degree of unorderedness and unpredictability. These properties are not generally associated to the actions of linear operators in vector spaces and in finite dimensional ones there is a reason for that: there are no transitive operators in these spaces. However, in infinite dimensional vector spaces there are operators with the same dynamical properties as the ones found in continuous and ergodic dynamics in non-linear compact metric spaces. Besides these properties, the Dynamical Systems theory also takes interest in whether interesting dynamical properties of a certain function are preserved under sufficiently small perturbations. Even though it is known that, in the linear context, this is not generally the case, it is possible to define an open class of operators where each one of them has rich dynamical properties when its domain is restricted.