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10 STATISTICAL DESCRIPTION OF ORBITS 72 Gauss map. of the form Any irrational real number x ∈ ]0, 1] has a unique continued fraction representation x = [0; a 1 , a 2 , a 3 , . . . ] = a 1 + 1 a 2 + 1 1 a 3 + 1 . .. where the a n are nonnegative integers. The equality sign and the “infinite fraction” above mean that the sequence of finite continued fractions 1 r n = [0; a 1 , a 2 , . . . , a n ] = 1 a 1 + 1 a 2 + . .. 1 + a n which are called “convergents”, do converge to x as n → ∞. The sequence of rationals r n (any such r n provide the best rational approximation for x with denominator less or equal than that of r n , as you may have been teached in a course on number theory) is inductively constructed as follows. First, observe that if a 1 = [1/x] and x 1 = 1/x − a 1 we may write 1 x = a 1 + x 1 with x 1 ∈ [0, 1]. Then, since x 1 ≠ 0, for otherwise x would be rational, we may define a 2 = [1/x 1 ] and x 2 = 1/x 1 − a 2 to get 1 x = 1 a 1 + a 2 + x 2 Inductively, we see that 1 x = 1 a 1 + 1 a 2 + ... + 1 a n+xn where x n = 1/x n−1 − a n and a n = [1/x n−1 ]. This amounts to say that the sequence (x n ) is the trajectory of x under the Gauss map g : ]0, 1] → ]0, 1], defined as x ↦→ 1/x − [1/x] Observe that g is not defined at the origin, hence to iterate g we need to avoid all the preimages of 0, which are the rationals. This is not a problem if we want to study the statistical properties of g with respect to Lebesgue measure, since rationals form a subset of zero measure. The Gauss map admits an absolutely continuous invariant measure µ = ρdx, defined as µ (A) = 1 log 2 · ∫ A 1 1 + x dx for any Borel subset A ⊂ ]0, 1]. The denominator log 2 is there to normalize the measure, so we just have to check the invariance criterium for the density ρ (x) = 1/ (1 + x). Since any x ′ ∈ ]0, 1] has one preimage x k = 1/ (x ′ + k) in each interval ]1/ (k + 1) , 1/k], we compute ∑ x∈g −1 {x ′ } ρ (x) |det g ′ (x)| x 2 k = ∑ 1 + x k k≥1 = ∑ ( ) 1 x ′ + k − 1 x ′ + k + 1 k≥1 = 1 1 + x ′ = ρ (x′ )
10 STATISTICAL DESCRIPTION OF ORBITS 73 and we are done.
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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,
Page 21 and 22: 4 NÚMEROS E DINÂMICA 21 4 Número
Page 23 and 24: 4 NÚMEROS E DINÂMICA 23 4.2 Deslo
Page 25 and 26: 4 NÚMEROS E DINÂMICA 25 Rotaçõe
Page 27 and 28: 4 NÚMEROS E DINÂMICA 27 Observe q
Page 29 and 30: 4 NÚMEROS E DINÂMICA 29 Frações
Page 31 and 32: 5 ÓRBITAS REGULARES E PERTURBAÇÕ
Page 39 and 40: 6 FLOWS 39 6 Flows “... forse sti
Page 41 and 42: 6 FLOWS 41 6.2 Integration of one-d
Page 43 and 44: 6 FLOWS 43 Counter-example. Both th
Page 45 and 46: 6 FLOWS 45 which undergoes a catast
Page 47 and 48: 6 FLOWS 47 Uniqueness of solutons.
Page 49 and 50: 6 FLOWS 49 Dependence on initial da
Page 51 and 52: 7 OSCILLATIONS AND CYCLES 51 7 Osci
Page 53 and 54: 7 OSCILLATIONS AND CYCLES 53 Phase
Page 55 and 56: 7 OSCILLATIONS AND CYCLES 55 The ge
Page 57 and 58: 7 OSCILLATIONS AND CYCLES 57 para a
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 71: 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
Page 81 and 82: 12 TRANSITIVIDADE E ÓRBITAS DENSAS
Page 83 and 84: 13 HOMEOMORFISMOS DO CÍRCULO 83 13
Page 85 and 86: 13 HOMEOMORFISMOS DO CÍRCULO 85 de
Page 87 and 88: 14 PERDA DE MEMÓRIA E INDEPENDÊNC
Page 97 and 98: 15 DIMENSIONS, FRACTALS AND ENTROPY
Page 103 and 104: 16 ERGODICITY AND CONVERGENCE OF TI
Page 105 and 106: 16 ERGODICITY AND CONVERGENCE OF TI
Page 107 and 108: REFERÊNCIAS 107 Referências [AA67
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