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Title:  Directed graph iterated function systems 
Authors:  Boore, Graeme C. 
Supervisors:  Falconer, K. J. Olsen, Lars 
Keywords:  Fractal geometry Iterated function systems Exact Hausdorff measure of attractors Directed graphs Multifractal analysis Qth packing moment 
Issue Date:  30Nov2011 
Abstract:  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 2vertex directed graph IFSs have attractors
that cannot be the attractors of standard (1vertex 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 selfsimilar measure at a vertex u, for the nonlattice 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 nonnegative
matrix associated with the system is irreducible.
In the nonlattice 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 boxdimension 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. 
URI:  http://hdl.handle.net/10023/2109 
Type:  Thesis 
Publisher:  University of St Andrews 
Appears in Collections:  Pure Mathematics Theses

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