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dc.contributor.advisorFraser, Jonathan M.
dc.contributor.advisorFalconer, K. J.
dc.contributor.authorBurrell, Stuart Andrew
dc.coverage.spatialviii, 116 p.en_US
dc.date.accessioned2021-06-17T16:01:37Z
dc.date.available2021-06-17T16:01:37Z
dc.date.issued2021-06-29
dc.identifier.urihttps://hdl.handle.net/10023/23381
dc.description.abstractWe investigate the dimension and structure of four fractal families: inhomogeneous attractors, fractal projections, fractional Brownian images, and elliptical polynomial spirals. For each family, particular attention is given to the relationships between different notions of dimension. This may take the form of determining conditions for them to coincide, or, in the case they differ, calculating the spectrum of dimensions interpolating between them. Material for this thesis is drawn from the papers [6,7,8,9,10]. First, we develop the dimension theory of inhomogeneous attractors for non-linear and affine iterated function systems. In both cases, we find natural quantities that bound the upper box-counting dimension from above and identify sufficient conditions for these bounds to be obtained. Our work improves and unifies previous theorems on inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer's seminal results on homogeneous self-affine sets. Second, we prove that the intermediate dimensions of the orthogonal projection of a Borel set 𝐸 ⸦ ℝⁿ onto a linear subspace 𝑉 are almost surely independent of the choice of subspace. Similar methods identify the almost sure value of the dimension of Borel sets under index-α fractional Brownian motion. Various applications are given, including a surprising result that relates the box dimension of the Hölder images of a set to the Hausdorff dimension of the preimages. Finally, we investigate fractal aspects of elliptical polynomial spirals; that is, planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis, computing explicitly their intermediate, box-counting and Assouad-type dimensions. Relying on this, we bound the Hölder regularity of maps that deform one spiral into another, generalising the `winding problem’ of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents.en_US
dc.description.sponsorship"This work was supported by a PhD scholarship from The Carnegie Trust for the Universities of Scotland [grant number EP/L1234567/8]." -- Fundingen
dc.language.isoenen_US
dc.publisherUniversity of St Andrews
dc.relationS. A. Burrell. On the dimension and measure of inhomogeneous attractors. Real Anal. Exchange, 44 (1), 199–216 (2019)en_US
dc.relationS. A. Burrell and J. M. Fraser. The dimensions of inhomogeneous self-affine sets. Ann. Acad. Sci. Fenn. Math., 45, 313–324 (2020)en_US
dc.relationS. A. Burrell. Dimensions of fractional Brownian images. Preprint (2020)en_US
dc.relationS. A. Burrell, K. J. Falconer and J. M. Fraser. The fractal structure of elliptical polynomial spirals. Preprint (2020)en_US
dc.relationS. A. Burrell, K. J. Falconer and J. M. Fraser. Projection theorems for intermediate dimensions. J. Fractal Geom., 8 (2), 95–116 (2021)en_US
dc.rightsCreative Commons Attribution-NonCommercial-NoDerivatives 4.0 International*
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/*
dc.subjectFractal geometryen_US
dc.subjectIntermediate dimensionsen_US
dc.subjectAssouad spectrumen_US
dc.subjectInhomogeneous attractorsen_US
dc.subjectFractional Brownian motionen_US
dc.subjectProjectionen_US
dc.subjectElliptical polynomial spiralsen_US
dc.subject.lccQA614.86B88
dc.subject.lcshFractalsen
dc.subject.lcshAttractors (Mathematics)en
dc.subject.lcshBrownian motion processesen
dc.titleCoincidence and disparity of fractal dimensionsen_US
dc.typeThesisen_US
dc.contributor.sponsorCarnegie Trust for the Universities of Scotlanden_US
dc.type.qualificationlevelDoctoralen_US
dc.type.qualificationnamePhD Doctor of Philosophyen_US
dc.publisher.institutionThe University of St Andrewsen_US
dc.identifier.grantnumberEP/L1234567/8en_US


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