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 1
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