Topological Growth Rates and Fractal Dimensions

Throughout this thesis, we observe close correlations between values of the topological growth rates and various other fractal indices. These observations are based on both analytic derivations and numerical computations of the relevant exponents. In this chapter we derive inequalities that relate our topological growth rates to existing scaling indices such as the box-counting dimension and the Besicovitch-Taylor exponent. Such relationships lead to a better understanding of the topological growth rates. The chapter has three sections. We start by giving definitions of box-counting dimension, fat fractal exponents and Besicovitch-Taylor index. These measures of fractal scaling have close connections with one another, and with the topological growth rates. Sections 5.3.1 to 5.3.3 examine the disconnectedness and discreteness indices, and for subsets of and . The most detailed results are for compact totally disconnected subsets of the line; these are given in Section 5.3.1. Such sets are defined in terms of countably many complementary open intervals. It is well known that the fractal dimension is related to the scaling of the lengths of these deleted intervals. We adapt this result to show that and are also related to this scaling. In Section 5.3.2 we study subsets of higher-dimensional spaces, and obtain simple inequalities involving , , and the box-counting dimension . We give examples in Section 5.3.3 to illustrate some of the cases for the inequalities of Sections 5.3.1 and 5.3.2. A consequence of the results in this chapter is that for zero measure Cantor subsets of