Aspectos dinâmicos de operadores lineares
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Data
2020-02-14
Autores
Alves, Fabricio Fernando
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Universidade Estadual Paulista (Unesp)
Resumo
Nessa tese, estudamos algumas propriedades dinâmicas no contexto de Dinâmica Linear em espaços de Banach de dimensão infinita, como transitividade por cadeia, transitividade por cadeia em média, sombreamento e sombreamento médio, entre outros. Vários resultados foram provados e muitas outras questões foram propostas. No que concerne à transitividade por cadeia, provamos que toda componente conexa do espectro de um operador transitivo por cadeia intersecta o círculo unitário. A recíproca desse resultado foi investigada em dimensão finita e para deslocamentos ponderados. Estabelecemos também critérios para que uma classe de deslocamentos ponderados unilaterais seja transitivo por cadeia. Quanto á transitividade por cadeia em média, provamos que a classe dos operadores transitivos por cadeia em média contém a classe dos operadores transitivos por cadeia e dos operadores hiperbólicos. Estudamos ainda a transitividade por cadeia para uma transformação esférica induzida por uma matriz inversível. Além disso, foi demonstrado que um operador hiperbólico possui sombreamento médio; que em dimensão finita hiperbolicidade e sombreamento médio coincidem; que, na presença de expansividade uniforme, a propriedade de sombreamento médio implica a de sombreamento. Introduzimos também o conceito de expansividade média e demonstramos que expansividade média mais sombreamento médio implicam expansividade uniforme e, em particular, hiperbolicidade.
In this thesis, we studied some dynamical properties on the context of Linear Dynamics on infinite-dimensional Banach spaces, such as chain transitivity, average chain transiti vity, shadowing and average shadowing, among others. Many results were proved and many other questions were proposed. Concerning chain transitivity, we proved that each connected component of the spectrum of a chain transitive operator intersects the unit circle. The converse was investigated for finite dimension and for weighted shifts. We also established criteria so that a class of unilateral weighted shifts may be chain transitive. For average chain transitivity, we proved that the class of average chain transitive ope rators contains the class of the chain transitive operators and the class of the hyperbolic ones. We studied yet the chain transitivity for the spherical transformation induced by an invertible matrix. Moreover, we showed that a hyperbolic operator has average shadowing property; in finite dimension, hyperbolicity and average shadowing coincide; under the presence of uniform expansivity, the average shadowing property implies the shadowing property. We also introduced the concept of average expansivity and then we proved that average expansivity plus average shadowing imply uniform expansivity and, in particular, hyperbolicity.
In this thesis, we studied some dynamical properties on the context of Linear Dynamics on infinite-dimensional Banach spaces, such as chain transitivity, average chain transiti vity, shadowing and average shadowing, among others. Many results were proved and many other questions were proposed. Concerning chain transitivity, we proved that each connected component of the spectrum of a chain transitive operator intersects the unit circle. The converse was investigated for finite dimension and for weighted shifts. We also established criteria so that a class of unilateral weighted shifts may be chain transitive. For average chain transitivity, we proved that the class of average chain transitive ope rators contains the class of the chain transitive operators and the class of the hyperbolic ones. We studied yet the chain transitivity for the spherical transformation induced by an invertible matrix. Moreover, we showed that a hyperbolic operator has average shadowing property; in finite dimension, hyperbolicity and average shadowing coincide; under the presence of uniform expansivity, the average shadowing property implies the shadowing property. We also introduced the concept of average expansivity and then we proved that average expansivity plus average shadowing imply uniform expansivity and, in particular, hyperbolicity.
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Dinâmica linear, Transitividade por cadeia, Transitividade por cadeia em média, Sombreamento, Sombreamento em média, Linear dynamics, Chain transitivity, Average chain transitivity, Shadowing, Average shadowing