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Master's Dissertation
DOI
10.11606/D.55.2018.tde-30102018-095453
Document
Author
Full name
Herbert Milton Ccalle Maquera
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2018
Supervisor
Committee
Tahzibi, Ali (President)
Lima, Yuri Gomes
Mencattini, Igor
Ponce, Gabriel
Title in Portuguese
Teorema de Furstenberg sobre o produto aleatório de matrizes
Keywords in Portuguese
Expoentes de Lyapunov
Medidas estacionarias
Produto aleatório de matrizes
Abstract in Portuguese
Nesta dissertação estudamos de um ponto de vista probabilístico, o comportamento assintótico de sistemas dinâmicos. Um exemplo simples de formular e profundo é o estudo de produto aleatório de matrizes (FURSTENBERG; KESTEN, 1960). Utilizaremos como ferramenta o estudo dos cociclos lineares, posteriormente mediante o Teorema de Furstenberg-Kesten definiremos o expoente de Lyapunov do cociclo, em seguida enunciamos e provamos o Teorema Ergódico Multiplicativo de Oseledets o qual nos permite entender o comportamento das órbitas típicas para um cociclo dado F : M x R2 → M x R2. O Teorema de Fusrtenberg-Kesten fornece informações sobre o crescimento das matrizes An(x), enquanto o Teorema de Oseledets descreve o comportamento assintótico dos vetores An(x).v. Finalmente provamos o teorema principal desta dissertação, o Teorema de Furstenberg o qual diz que na maioria dos casos o maior expoente de Lyapunov é positivo (FURSTENBERG, 1963).
Title in English
Furstenberg theorem on the random product of matrices
Keywords in English
Lyapunov exponent
Random product of matrices
Stationary measures
Abstract in English
In this thesis we study from a probabilistic point of view, the asymptotic behavior of dynamic systems, a deep and simple example is the random product of matrices (FURSTENBERG; KESTEN, 1960). We will use as a tool, the study of linear cocycles, later using the Furstenberg- Kesten Theorem we will define the Lyapunov exponent of the cocycle, then we enunciate and prove the Multiplicative Ergodic Theorem of Oseledets which allow us to understand the behavior of the typical orbits for a given cocycle F : M x R2 → M x R2. The Fusrtenberg- Kesten theorem provides information on the growth of the matrices A(x), while the theorems of Oseledets describe the asymptotic behavior of the vectors An(x).v. Finally we prove our main theorem, Furstenbergs Theorem which states that in most cases the greatest exponent of Lyapunov is positive (FURSTENBERG, 1963).
 
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Publishing Date
2018-10-30
 
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