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MAC850 – Teoria Ergódica Diferenciável
Ementa: Expoentes de Lyapunov | Teorema de Oseledets e desigualdade de Ruelle | Teoria de hiperbolicidade não uniforme (de Pesin) e propriedades de medidas hiperbólicas | Fórmula de Pesin | Ergodicidade do fluxo geodésico em superfícies de curvatura constante negativa | Medidas hiperbólicas e Teorema de Katok | Atratores e medidas físicas.
Lista de Exercícios:
Lista 1 (corrigido, com gabarito parcial)
Seminários:
Princípio variacional para sequências de funções sub-additivas ([Cao, Feng, Huan]) (Lamartine, 24/4)
Brin-Katok's formula of entropy (Matheus, 15/5)
Entropia-expoente-dimensão de medidas ergódicas ([Young]) (Talita, 12/6)
Dimension approximation in smooth dynamics (Harold, 12/6)
Analytic dependence of Lyapunov exponents on transition probabilities (Artur, 5/6)
Medidas físicas e atratores não-uniformemente hiperbólicos (Thiago, 26/6 3/7)
Lp-genereic cocycles have one-point Lyapunov spectrum (João Victor, 29/5)
Semi-continuidade de entropia (Enos, 19/6)
Critérios para avaliação de um seminário
Bibliografia: (os lívros (*) estão disponíveis na Plataforma CAPES)
[BP] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynam- ics of Systems with Nonzero Lyapunov Exponents, Cambridge, 2007.
[CK] Climenhaga, Katok, Measure theory through dynamical eyes, arXiv:1208.4550
[ABMORST] Brown, Alvarez, Malicet, Obata, Roldán, Santiago, Triestino, Entropy, Lyapunov exponents, and rigidity of group actions, versão final: Ensaios Matemáticos SBM
[M] (*) Mañé - Ergodic Theory and Differentiable Dynamics. Springer-Verlag, New-York, 1987
[KH] Katok, Hasselblatt - Introduction to the Modern Theory of Dynamical Systems, Cambridge, 1995
[OV] Oliveira, Viana, Fundamentos da Teoria Ergódica, Sociedade Brasileira de Matemática, Coleção Fronteiras da Matemática, 2014.
[W] Walters, Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982
[V] Viana, Lectures on Lyapunov exponents. Cambridge Studies in Advanced Mathematics, 145. Cambridge University Press, Cambridge, 2014.
[P] Pollicott, Lectures on ergodic theory and Pesin theory on compact manifolds. London Mathematical Society Lecture Note Series, 180. Cambridge University Press, Cambridge, 1993.
[M] Merry, Lecture Notes Dynamical Systems and Ergodic Theory
Notas do curso (atualizadas dia 24/04)
Slides medidas hiperbólicas & mudança de norma
Slides finer structures of hyperbolic measures
Slides Solenoid (and its exponents)
Demais referências:
[AB] Avila, Bochi, On the subadditive ergodic theorem, ver página de Jairo Bochi
[FHY] Fathi, Herman, Yoccoz, A proof of Pesin's stable manifold theorem.
[FK] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960), 457–469.
[K] Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B 30 (1968), 499–510.
[O] V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197–231.
[R2] D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. Math. Inst. Hautes Études Sci. 50 (1979), 27–58.
[H] Heuser, Lehrbuch der Analysis, Teil 2, B.G.Teubner, Kapitel XXIV (para formas multilineares!)
[AMR] Abraham, Marsden, Ratiu, Manifolds, Tensor Analysis, and Applications, Springer, Chapter 6 (para formas multilineares!)
[T] Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, Chapter V (para formas multilineares!)
[Y] Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), 109-124.
[Y2] Young, Some large deviation results for dynamical systems, Trans. Amer. Math. Soc. 318 (1990), 525-543.
[BK] Brin, Katok, On local entropy, Geometric dynamics, Proc. int. Symp., Rio de Janeiro/Brasil 1981, Lect. Notes Math. 1007, 30-38 (1983).
[Ka] Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math., Inst. Hautes Étud. Sci. 51, 137-173 (1980).
[G] Gelfert, Repellers for non-uniformly expanding maps with singular or critical points. Bull. Braz. Math. Soc. (N.S.) 41, No. 2, 237-257 (2010).
exemplos de box counting <> Hausdorff dimension: McMullen, Curt, The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96, 1-9 (1984).