CHAPTER 1

Introduction to Fractals

Unlike many other mathematical notions, the notion of a fractal is not really

well-defined by some axiomatic list of properties. The best way to understand what

we mean by a fractal is to contrast fractals with classical geometry objects like lines

and surfaces. The latter are usually studied using tools from Differential Geometry

and, ultimately, Linear Algebra. They are linear objects, which are locally well

approximated by their tangent planes. This linearity is absent when we deal with

fractals.

To formalize this concept of non-linearity , we will introduce the notions of

mini- and micro-sets. We will start with the definition of the Hausdorff distance

between subsets of a metric space.

Definition. Let X be a compact metric space and let σ(X) be the set of all

its closed subsets. If x ∈ X and A ∈ σ(X), we define the distance from x to A as

the minimum of all distances from x to points of A, i.e.,

d(x, A) = min

y∈A

d(x, y).

x

y

A

Figure 1. The distance from a point x to a set A is the distance

from x to a nearest point y ∈ A.

If A, B ∈ σ(X) are two closed subsets of X, we define the distance between

them as

D(A, B) = max{max

x∈A

d(x, B), max

y∈B

d(y, A)}.

Thus, the inequality D(A, B) ε means that A is contained in the ε-neighbor-

hood of B and B is contained in the ε-neighborhood of A, i.e., for every x ∈ A,

there exists y ∈ B such that d(x, y) ε and vice versa.

Note that the sets close in this metric may be of very different nature from the

topological perspective. For instance, the interval [0, 1] is ε-close to the finite set

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http://dx.doi.org/10.1090/cbms/120/01