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In this thesis we study actions of countable groups on measure spaces underthe assumption that the dynamics are non-singular, with particular reference topointwise ergodic theorems and their relationship to the critical dimensions ofthe action.

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The Multiplicative Ergodic Theorem is a powerful tool for studying certain types of dynamical systems, involving real matrix cocycles. It gives a block diagonalization of these cocycles, according to the Lyapunov exponents. We ask if it is always possible to refine the diagonalization to a block upper-triangularization, and if not over the real numbers, then over the complex numbers. After building up to the posing of the question, we prove that there are counterexamples to this statement, and give concrete examples of matrix cocycles which cannot be block upper-triangularized. / Graduate / 0405 / jahoran@uvic.ca

Orientador: Eduardo Garibaldi / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T22:00:53Z (GMT). No. of bitstreams: 1 Gomes_JoaoTiagoAssuncao_M.pdf: 1424859 bytes, checksum: 6679c0458ae1350513c30bdc7ea2e699 (MD5) Previous issue date: 2012 / Resumo: Exibir os estados de Gibbs e os estados de equilíbrios para certos sistemas de spins sobre reticulados é um problema de grande interesse para mecânica estatística. Com este intuito, apresentamos para o caso unidimensional dois formalismos existentes para tais sistemas: o formalismo DLR (enfoque mecânico-estatístico) e o formalismo SRB (enfoque dinamicista). Apesar das particularidades próprias aos contextos nos quais cada um dos formalismos se aplica, investigam-se aqui as relações existentes entre estes através da energia livre de Gibbs e da pressão topológica. Discute-se também o comportamento assintótico dos estados de Gibbs/equilíbrio quando levados ao congelamento do sistema. Tal fenômeno nos conduz ao estudo dos estados maximizantes via teoria de otimização ergódica. Ao fim, comparam-se algumas ideias da álgebra max/min-plus e o conceito de subação, as quais serão fundamentais para análise do comportamento assintótico da pressão topológica / Abstract: To exhibit Gibbs states and equilibrium states for certain kind of lattice spin systems is a problem with great interest for statistical mechanics. To that end, we introduce two existing formalisms for one-dimensional systems: DLR formalism (statistical-mechanical approach) and SRB formalism (dynamical-systems approach). In spite of their distinct applications, we analyse the relation between them through the notions of Gibbs free energy and topological pressure. We discuss also the asymptotic behaviour of Gibbs/equilibrium states when the system is frozen. This phenomenon leads us to the study of maximizing states in the context of ergodic optimization. Finally, we compare some ideas of max/min-plus algebra and the notion of sub-action, which will be essential to investigate the asymptotic behaviour of the topological pressure / Mestrado / Matematica / Mestre em Matemática

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We apply anisotropic Banach space methods together with operator renewal theory to obtain polynomial rates of decay of correlations for a class of generalized baker's transformations. The polynomial rates were proved for a smaller class of observables in a 2013 paper of Bose and Murray by fundamentally different methods. Our approach provides a direct analysis of the Frobenius-Perron operator associated to a generalized baker's transformation in contrast to the paper of Bose and Murray where decay rates are obtained for a factor map and lifted to the full map. / Graduate

Demonstramos o seguinte teorema: Seja M uma variedade Riemanniana compacta, conexa e sem bordo. Dados um endomorismo f : M ightarrow M, uma função contínua \\phi: M ightarrow R e \\epsilon > 0, então existe um endomorísmo \\tilde f : M ightarrow M tal que d(f; \\tide f) = \\max_{x \\in M} d(f(x); \\tilde f(x)) < \\epsilon, e existe uma medida \\phi-maximizante para \\tilde f que está suportada em uma orbita periodica. Este teorema e uma generalização dos resultados obtidos por S. Addas-Zanatta e F. Tal. / We prove the following theorem: Let M be a bondaryless, compact and connected Riemannian Manifold. Given an endomorphism f on M, a continuous function \\phi : M ightarrow R and \\epsilon > 0, then there exist an endomorphism \\tilde f on M with d(f; \\tilde f) < \\epsilon such that, some \\phi-maximizing measure for \\tilde f is supported on a periodic orbit. This theorem is a generalization of the results obtained by S. Addas-Zanatta and F. Tal.

Nessa dissertação estudamos Sistemas Dinâmicos do ponto de vista da Otimização Ergódica. Analizamos o problema da maximização da integral de potenciais com respeito a probabilidades invariantes pela dinâmica. Mostramos que toda medida ergódica e unicamente maximizante para algum potencial. Verificamos que o conjunto de potenciais com exatamente uma medida maximizadora e residual. Esses resultados são obtidos atrav es de técnicas da Teoria Ergódica e Análise Convexa. / In this thesis we study dynamical systems trough the viewpoint of ergodic optimization. We analyze the problem of maximizing integrals of potentials with respect to invariant probabilities. We show that every ergodic measure is uniquely maximizing for some potential. We also verify that the set of potentials with exactly one maximizing measure is residual. This results are obtained through techniques of ergodic theory and convex analysis.