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dc.contributor.authorFraser, Jonathan M.
dc.contributor.authorYu, Han
dc.date.accessioned2019-02-23T00:33:29Z
dc.date.available2019-02-23T00:33:29Z
dc.date.issued2018-04-30
dc.identifier.citationFraser , J M & Yu , H 2018 , ' New dimension spectra : finer information on scaling and homogeneity ' , Advances in Mathematics , vol. 329 , pp. 273-328 . https://doi.org/10.1016/j.aim.2017.12.019en
dc.identifier.issn0001-8708
dc.identifier.otherPURE: 251802899
dc.identifier.otherPURE UUID: 1b7b3400-e5a7-49b0-a82b-46e170ec06b6
dc.identifier.otherScopus: 85042326730
dc.identifier.otherORCID: /0000-0002-8066-9120/work/58285476
dc.identifier.otherWOS: 000431089100008
dc.identifier.urihttps://hdl.handle.net/10023/17146
dc.descriptionFunding: Leverhulme Trust Research Fellowship (RF-2016-500) (JMF).en
dc.description.abstractWe introduce a new dimension spectrum motivated by the Assouad dimension; a familiar notion of dimension which, for a given metric space, returns the minimal exponent α ≥ 0 such that for any pair of scales 0 < r < R , any ball of radius R may be covered by a constant times (R/r)α balls of radius r. To each θ ∈ (0,1), we associate the appropriate analogue of the Assouad dimension with the restriction that the two scales r and R used in the definition satisfy log R/ log r=θ. The resulting ‘dimension spectrum’ (as a function of θ) thus gives finer geometric information regarding the scaling structure of the space and, in some precise sense, interpolates between the upper box dimension and the Assouad dimension. This latter point is particularly useful because the spectrum is generally better behaved than the Assouad dimension. We also consider the corresponding ‘lower spectrum’, motivated by the lower dimension, which acts as a dual to the Assouad spectrum. We conduct a detailed study of these dimension spectra; including analytic, geometric, and measureability properties. We also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with sub-exponential and monotonic winding. We also give several applications of our results, including: dimension distortion estimates under bi-Hölder maps for Assouad dimension and the provision of new bi-Lipschitz invariants.
dc.format.extent56
dc.language.isoeng
dc.relation.ispartofAdvances in Mathematicsen
dc.rights© 2018 Elsevier Inc. All rights reserved. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.aim.2017.12.019en
dc.subjectAssouad dimensionen
dc.subjectLower dimensionen
dc.subjectBox-counting dimensionen
dc.subjectContinuityen
dc.subjectMeasureabilityen
dc.subjectUnwinding spiralsen
dc.subjectQA Mathematicsen
dc.subjectT-NDASen
dc.subjectBDCen
dc.subjectR2Cen
dc.subject.lccQAen
dc.titleNew dimension spectra : finer information on scaling and homogeneityen
dc.typeJournal articleen
dc.contributor.sponsorThe Leverhulme Trusten
dc.description.versionPostprinten
dc.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.identifier.doihttps://doi.org/10.1016/j.aim.2017.12.019
dc.description.statusPeer revieweden
dc.date.embargoedUntil2019-02-23
dc.identifier.grantnumberRF-2016-500en


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