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dc.contributor.advisorFalconer, K. J.
dc.contributor.advisorOlsen, Lars
dc.contributor.authorBoore, Graeme C.
dc.description.abstractThis 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 attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or 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 Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing 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-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μᵤ to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μᵤ, of the set consisting of all the intersections of the components of Fᵤ, is strictly less than the multifractal q Hausdorff dimension with respect to μᵤ of Fᵤ.en_US
dc.publisherUniversity of St Andrews
dc.subjectFractal geometryen_US
dc.subjectIterated function systemsen_US
dc.subjectExact Hausdorff measure of attractorsen_US
dc.subjectDirected graphsen_US
dc.subjectMultifractal analysisen_US
dc.subjectQth packing momenten_US
dc.subject.lcshHausdorff measuresen_US
dc.subject.lcshDirected graphsen_US
dc.subject.lcshIterative methods (Mathematics)en_US
dc.titleDirected graph iterated function systemsen_US
dc.contributor.sponsorEngineering and Physical Sciences Research Council (EPSRC)en_US
dc.type.qualificationnamePhD Doctor of Philosophyen_US
dc.publisher.institutionThe University of St Andrewsen_US

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