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dc.contributor.advisorFraser, Jonathan M.
dc.contributor.advisorFalconer, K. J.
dc.contributor.authorBanaji, Amlan
dc.coverage.spatialvii, 174 p.en_US
dc.description.abstractHausdorff and box dimension are two familiar notions of fractal dimension. Box dimension can be larger than Hausdorff dimension, because in the definition of box dimension, all sets in the cover have the same diameter, but for Hausdorff dimension there is no such restriction. This thesis focuses on a family of dimensions parameterised by θ ∈ (0,1), called the intermediate dimensions, which are defined by requiring that diam(U) ⩽ (diam(V))ᶿ for all sets U, V in the cover. We begin by generalising the intermediate dimensions to allow for greater refinement in how the relative sizes of the covering sets are restricted. These new dimensions can recover the interpolation between Hausdorff and box dimension for compact sets whose intermediate dimensions do not tend to the Hausdorff dimension as θ → 0. We also use a Moran set construction to prove a necessary and sufficient condition, in terms of Dini derivatives, for a given function to be realised as the intermediate dimensions of a set. We proceed to prove that the intermediate dimensions of limit sets of infinite conformal iterated function systems are given by the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This applies to sets defined using continued fraction expansions, and has applications to dimensions of projections, fractional Brownian images, and general Hölder images. Finally, we determine a formula for the intermediate dimensions of all self-affine Bedford–McMullen carpets. The functions display features not witnessed in previous examples, such as having countably many phase transitions. We deduce that two carpets have equal intermediate dimensions if and only if the multifractal spectra of the corresponding uniform Bernoulli measures coincide. This shows that if two carpets are bi-Lipschitz equivalent then the multifractal spectra are equal.en_US
dc.description.sponsorship"This work was supported by a Leverhulme Trust Research Project Grant (RPG-2019-034)." -- Fundingen
dc.rightsCreative Commons Attribution-NonCommercial-NoDerivatives 4.0 International*
dc.subjectFractal geometryen_US
dc.subjectHausdorff dimensionen_US
dc.subjectBox dimensionen_US
dc.subjectIntermediate dimensionsen_US
dc.subjectDimension interpolationen_US
dc.subjectMoran seten_US
dc.subjectIterated function systemen_US
dc.subjectBedford–McMullen carpeten_US
dc.subject.lcshDimension theory (Topology)en
dc.subject.lcshMathematical analysisen
dc.titleInterpolating between Hausdorff and box dimensionen_US
dc.contributor.sponsorLeverhulme Trusten_US
dc.type.qualificationnamePhD Doctor of Philosophyen_US
dc.publisher.institutionThe University of St Andrewsen_US

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    Except where otherwise noted within the work, this item's licence for re-use is described as Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International