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15 DIMENSIONS, FRACTALS AND ENTROPY 96 15 Dimensions, fractals and entropy 15.1 Dimensions of metric spaces Conjunto ε-densos e ε-separados. The following notions of the approximate massiveness of a metric space (X, d) are due to Kolmogorov’s school 29 . An ε-covering of (X, d) is a covering of X ⊂ ∪ α C α by subsets of diameters diam(C α ) < 2ε. Call C ε (X, d) the minimal cardinality of an ε-covering of X. An ε-net for (X, d) is a collection A ⊂ X of points such that any point of X is at a distance not exceeding ε from some point of A, i.e. X ⊂ ∪ a∈A {x ∈ X s.t. d(x, a) < ε}. Call N ε (X, d) the minimal cardinality of an ε-net for X. If X is a centered space (any subset of diameter 2r is contained in a ball of radius r centered in some point of X) then N ε (X, d) = C ε (X, d). A subset B ⊂ X is said ε-separated if its points are a distance greater than ε from each other. Call S ε (X, d) the maximal cardinality of a set of ε-separated points inside X. These three definitions make sense if the above extremal cardinalities are finite for every ε > 0, and it is not difficult to see that this happens simultaneously. The class of metric spaces with this property is called the class of totally bounded sets and the main examples are compact spaces. The base 2 logarithms of these quantities have interpretations related to the probabilistic theory of transmission of signal, and are called Exercícios. log 2 C ε (X, d) minimal (or absolute) ε-entropy of (X, d) log 2 N ε (X, d) ε-entropy of (X, d) log 2 S ε (X, d) ε-capacity of (X, d) • Show that an ε-net defines an ε-covering, and any ε-covering determines a 2ε-net, so that C ε (X, d) ≤ N ε (X, d) ≤ C 2ε (X, d) • Show that a maximal ε-separated set is a ε-net, and that any ε-ball centered at a point of a minimal ε-net cannot contain more than one point of a 2ε-separated set, so that S 2ε (X, d) ≤ N ε (X, d) ≤ S ε (X, d) Box-counting dimensions. The upper and lower box counting dimension (also known as Minkowski dimensions or metric dimensions) of the metric space (X, d) are defined as dim b (X) = lim sup ε↓0 − log N ε(X, d) log ε dim b (X) = lim inf ε↓0 − log N ε(X, d) log ε We get the same values if we substitute S ε (X, d) or C ε (X, d) to N ε (X) in the above formulas (just compare the counting functions at the values ε and 2ε). For reasonable self-similar metric spaces the two limits coincide, and their common value dim b (X) is simply called box counting dimension. Exercícios. • Show that the box-counting dimension of the n-dimensional cube [0, 1] n is what you expect, namely dim b ([0, 1] n ) = n. • Consider the interval [0, 1] equipped with the Euclidean metric d, and define new metrics d α (x, y) = d(x, y) α , for α ≤ 1. Compute the box-counting dimension of ([0, 1], d α ). 29 A.N. Kolmogorov and V.M. Tihomirov, ε-entropy and ε-capacity of sets in functional spaces, Uspekhi Mat. Nauk 14 (1959), 3-86. [Translated in Amer. Math. Soc. Transl., series 2, 17 (1961), 277-364.]
15 DIMENSIONS, FRACTALS AND ENTROPY 97 15.2 Fractals The word “fractal” (from the Latin FRACTUS) was coined by Benoît Mandelbrot in 1975 [Ma75] to generically denote a family of self-similar, scale-invariant metric spaces with non-integer boxcounting (or other) dimension. Peano curves. The dimension is not preserved under continuous maps! In 1890 Giuseppe Peano 30 discovered the existence of “space-filling curves”, continuous maps of the interval [0, 1] onto the unit square [0, 1] × [0, 1]. Here is a modern construction. The middle-third Cantor set K ⊂ [0, 1] is homeomorphic to its powers, in particular to K × K. Since there exists a continuous map of K onto [0, 1], one can define a continuous map f of K onto the unit square [0, 1] × [0, 1]. This map can be extended (the complement [0, 1]\K is a countable union of open intervals (a n , b n ), where we may take, for example, the affine segment joining f(a) to f(b)) to a continuous map of the unit interval onto the unit square. Middle-third Cantor set and devil’s staircase. Seven stages of the construction of the Cantor set and graph of the Cantor function (from http://en.wikipedia.org/wiki/Cantor_set and http://en.wikipedia.org/wiki/Cantor_function) Sierpinski gasket. Five stages of the construction of the Sierpinski gasket (from http://en.wikipedia.org/wiki/Sierpinski_gasket) von Koch snowflake. The von Koch snowflake is a continuous nowhere differentiable curve 31 . The box-counting dimension is log 4/ log 3. Four stages of the construction of the von Koch snowflake (from http://en.wikipedia.org/wiki/Koch_snowflake) 30 G. Peano, Sur une courbe, qui remplit toute une aire plane”, Mathematische Annalen 36 (1) (1890), 157-160. doi: 10.1007/BF01199438 31 Helge von Koch, Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire, Archiv för Matemat., Astron. och Fys. 1 (1904), 681- 702.
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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 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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