8 - Miscellaneous examples of fractal sets

Summary

Introduction

This chapter surveys examples of sets of fractional dimension which result from particular constructions or occur in other branches of mathematics or physics and relates them to earlier parts of the book. The topics have been chosen very much at the author's whim rather than because they represent the most important occurrences of fractal sets. In each section selected results of interest are proved and others are cited. It is hoped that this approach will encourage the reader to follow up some of these topics in greater depth elsewhere. Most of the examples come from areas of mathematics which have a vast literature; therefore in this chapter references are given only to the principal sources and to recent papers and books which contain further surveys and references.

Curves of fractional dimension

In this section we work in the (x,y)-coordinate plane and investigate the Hausdorff dimension of Γ, the set of points (x,f(x)) forming the graph of a function f defined, say, on the unit interval.

If f is a function of bounded variation, that is, if is bounded for all dissections 0 = x₀ < x₁ < … < x_m = l, then we are effectively back in the situation of Section 3.2; Γ is a rectifiable curve and so a regular 1-set. However, if f is a sufficiently irregular, though continuous, function it is possible for Γ to have dimension greater than 1. In such cases it can be hard to calculate the Hausdorff dimension and measure of Γ from a knowledge of f. However, if f satisfies a Lipschitz condition it is easy to obtain an upper bound.