This thesis concerns an active research area within fractal geometry.In the first part, in Chapters 2 and 3, for directed graph iterated function systems(IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractorsthat cannot be the attractors of standard (1-vertex directed graph) IFSs, withor without separation conditions. We also calculate their exact Hausdorff measure.Thus we are able to identify a new class of attractors for which the exact Hausdorffmeasure is known.We give a constructive algorithm for calculating the set of gap lengths of anyattractor as a finite union of cosets of finitely generated semigroups of positive realnumbers. The generators of these semigroups are contracting similarity ratios ofsimple cycles in the directed graph. The algorithm works for any IFS defined on ℝwith no limit on the number of vertices in the directed graph, provided a separationcondition holds.The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . Weobtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qthpacking moment of μᵤ, the self-similar measure at a vertex u, for the non-lattice case,with a corresponding limit for the lattice case. We do this(i) for any q ∈ ℝ if the strong separation condition holds,(ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negativematrix associated with the system is irreducible.In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrumof μᵤ to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the uppermultifractal q box-dimension with respect to μᵤ, of the set consisting of all the intersectionsof the components of Fᵤ, is strictly less than the multifractal q Hausdorffdimension with respect to μᵤ of Fᵤ.
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