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16 ERGODICITY AND CONVERGENCE OF TIME MEANS 106 theorem owes its name to the fact that 1 n + 1 n∑ ∫ ϕ (x + jα) → j=0 ϕdl uniformly for any continuous function ϕ on the circle, and this is interpreted as saying that the sequence of points {x, x + α, x + 2α, x + 3α, ...} is “equidistributed” w.r.t. Lebesgue measure. Linear flows on tori. Now, consider the torus X = R n /Z n of dimension n ≥ 2, and the linear flow φ t : x + Z n ↦→ x + tα + Z n defined by the differential equation ẋ = α where α ∈ R n . The “frequency vector” α = (α 1 , α 2 , ..., α n ) is said non resonant if the scalar product 〈α, k〉 = ∑ n j=1 α jk j ≠ 0 for any k ∈ Z n \ {0}. As above, one can approximate any continuous function on the torus with trigonometric functions. One then checks that ∫ 1 T ( ) e i2π〈k,x+tα〉 dt = ei2π〈k,x〉 e i2πT 〈k,x〉 − 1 → 0 T 0 iT 〈k, x〉 as T → ∞, for any k ∈ Z n \ {0}, while the time mean of the observable 1 is constant and equal to one. There follows that a non resonant linear flow on the torus is uniquely ergodic w.r.t. to Lebesgue measure. Dyadic adding machine. to be done
REFERÊNCIAS 107 Referências [AA67] V.I. Arnold & A. Avez, Problèmes ergodiques de la mécanique classique, Gauthier- Villars, 1967. [Ap69] T.M. Apostol, Calculus, John Wiley & Sons, New York 1969. [Ar78] [Ar79] V.I. Arnold, Metodi geometrici della teoria delle equazioni differenziali ordinarie, Editori Riuniti - MIR, Roma 1978. V.I. Arnold, Metodi matematici della meccanica classica, Edizioni MIR - Editori Riuniti, Roma 1979. [Ar85] V.I. Arnold, Equações diferenciais ordinárias, MIR 1985. [AS64] [BN05] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, 1964. P. Buttà e P. Negrini, Note del corso di Sistemi Dinamici, Università di Roma “La Sapienza”, 2005. [CG93] L. Carleson and T.W. Gamelin Complex dynamics, UTX, Springer-Verlag, 1993. [Chaos] P. Cvitanović, R. Artuso, P. Dahlqvist, R. Mainieri, G. Tanner, G. Vattay, N. Whelan and A. Wirzba, Chaos: Classical and Quantum, http://ChaosBook.org (Niels Bohr Institute, Copenhagen 2008). [CR48] R. Courant and H. Robbins, What is mathematics, Oxford University Press, 1948. [De89] R.L. Devaney, An introduction to chaotic dynamical systems, Addison-Wesley, 1989. [De92] R.L. Devaney, A first course in chaotic dynamical systems, Addison-Wesley, 1992. [Fa85] K. J. Falconer, The geometry of fractal sets, Cambridge University Press, 1985. [Fe63] [Gh07] [HK03] [HS74] [HSD04] [HW59] [Kh35] [KH95] [Ma75] [Mat95] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman lectures on physics, Addison- Wesley, Reading, 1963. E. Ghys, Résonances et petits diviseurs, in L’héritage scientifique de Kolmogorov, Berlin 2007. B. Hasselblatt and A. Katok, A first course in dynamics: with a panorama of recent developments, Cambridge University Press 2003. M.W. Hirsch and S. Smale, Differential equations, dynamical systems and linear algebra, Academic Press (Pure and Applied Mathematics. A series of Monographs and Textbooks), San Diego 1974. M.W. Hirsch, S. Smale and R.L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2nd ed., Elsevier Academic Press, 2004. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, fourth edition, Oxford University Press 1959. A.Ya. Khinchin, Continued Fractions, 1935 [translation by University of Chicago Press, 1954]. A. Katok and B. Hasselblat, Introduction to the modern theory of dynamical systems, Encyclopedia of mathematics and its applications, Cambridge University Press 1995. B. Mandelbrot, Les object fractals: forme, hasard, et dimension, Flammarion, Paris 1975. P. Mattila Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability, Cambridge University Press, 1995.
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1405R1 - Tópicos de Sistemas Dinâ
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CONTEÚDO 3 9 Transversalidade e bi
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1 INTRODUÇÃO 5 Hidrodinâmica. O
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2 ITERATION/RECURSION 7 2 Iteration
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2 ITERATION/RECURSION 9 do primeiro
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2 ITERATION/RECURSION 11 If a is th
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2 ITERATION/RECURSION 13 Nice pictu
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2 ITERATION/RECURSION 15 Experiênc
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3 SISTEMAS DINÂMICOS TOPOLÓGICOS,
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3 SISTEMAS DINÂMICOS TOPOLÓGICOS,
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4 NÚMEROS E DINÂMICA 21 4 Número
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4 NÚMEROS E DINÂMICA 23 4.2 Deslo
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4 NÚMEROS E DINÂMICA 25 Rotaçõe
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4 NÚMEROS E DINÂMICA 27 Observe q
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4 NÚMEROS E DINÂMICA 29 Frações
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5 ÓRBITAS REGULARES E PERTURBAÇÕ
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6 FLOWS 39 6 Flows “... forse sti
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6 FLOWS 41 6.2 Integration of one-d
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6 FLOWS 43 Counter-example. Both th
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6 FLOWS 45 which undergoes a catast
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6 FLOWS 47 Uniqueness of solutons.
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6 FLOWS 49 Dependence on initial da
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7 OSCILLATIONS AND CYCLES 51 7 Osci
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7 OSCILLATIONS AND CYCLES 53 Phase
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Page 59 and 60: 8 LINEARIZAÇÃO 59 8 Linearizaçã
Page 61 and 62: 9 TRANSVERSALIDADE E BIFURCAÇÕES
Page 63 and 64: 10 STATISTICAL DESCRIPTION OF ORBIT
Page 75 and 76: 11 RECORRÊNCIAS 75 Exercícios.
Page 77 and 78: 11 RECORRÊNCIAS 77 O conjunto não
Page 79 and 80: 12 TRANSITIVIDADE E ÓRBITAS DENSAS
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Page 83 and 84: 13 HOMEOMORFISMOS DO CÍRCULO 83 13
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Page 97 and 98: 15 DIMENSIONS, FRACTALS AND ENTROPY
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