My title - Departamento de Matemática da Universidade do Minho

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.]