This thesis considers a number of problems which are related to the study of random fractals. We define a class of iterations (which we call random hierarchical systems) of probability distributions, which are defined by applying a random map to a set of k independent and identically distributed random variables. Classical examples of this sort of iteration include the Strong Law of Large Numbers, Galton-Watson branching processes, and the construction of random self-similar sets. In Chapter 2, we consider random hierarchical systems on ℝ, under the condition that the random map is bounded above by a random weighted mean, and that the initial distribution is bounded below. Under moment conditions on the initial distribution we show that there exists almost sure convergence to a constant. In Chapters 3 to 5 we consider the asymptotics of some examples of random hierarchical systems, some of which arise when considering certain properties of random fractal graphs. In one example, which is related to first-passage percolation on a random hierarchical lattice, we show the existence of a family of non-degenerate fixed points and show that the sequence of distributions will converge to one of these. The results of some simulations are reported in Chapter 7. Part III investigates the spectral properties of random fractal graphs. In Chapter 8 we look at one example in detail, showing that there exist localised eigenfunctions which lead to certain eigenvalues having very high multiplicity. We also investigate the behaviour of the Cheeger constants of this example. We then consider eigenvalues of homogeneous random fractal graphs, which preserve some of the symmetry of deterministic fractals. We then use relationships between homogeneous graphs and more general random fractal graphs to obtain results on the eigenvalues of the latter. Finally, we consider a few further examples.

The use of fractals in fields such as molecular biology, epidemiology, landscape, ecology, geology, physics, etc., is becoming more common. In order to use fractals to model many phenomena, the researcher requires the knowledge of the fractal, or Hausdorff-Besicovitch, dimension. However, no statistical properties of the usual estimator, the entropy estimator, are known. In addition, the entropy estimator is biased high when an inefficient net is used. This dissertation develops a new estimator, the relative entropy estimator, which is asymptotically unbiased and is consistent. The estimator is asymptotically normal, and asymptotic confidence intervals are presented. An estimate of the variance of the estimator is given which does not depend on the dimension, or its estimate, using an occupancy model. The exact distribution of the estimator is also derived. Applications of the theory to various fields are presented. For example, I find that from the point of view of dimension, the logarithms of stock prices behave consistently with the classical Brownian function. Also, the relative entropy estimator gives a more realistic estimate of the dimension of surface terrain than an ad hoc estimate found in the literature. The Hausdorff dimensions of nursery-grown tree roots were estimated, and it was found that the dimension is related to the probability of the tree’s survival when the tree is planted in the wild. The dimensions of Julia sets and of the Hénon attractor were also investigated. A computer program for calculating the estimates is included. / Ph. D.