Show simple item record

Files in this item

Thumbnail

Item metadata

dc.contributor.advisorOlsen, Lars
dc.contributor.authorSnigireva, Nina
dc.coverage.spatial111en
dc.date.accessioned2009-05-01T13:17:19Z
dc.date.available2009-05-01T13:17:19Z
dc.date.issued2008
dc.identifier.urihttp://hdl.handle.net/10023/682
dc.description.abstractThe thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and measures. In particular, we show that these sets and measures are natural generalizations of the well known self-similar sets and measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary) self-similar sets. In the third chapter we investigate the L^{q} spectra and the Renyi dimensions of inhomogeneous self-similar measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case. Namely, we show that inhomogeneous self-similar measures may have phase transitions which is in sharp contrast to the behaviour of the L^{q} spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra of inhomogeneous self-similar measures. We show that the multifractal spectra of inhomogeneous self-similar measures may be non-concave which is again in sharp contrast to the behaviour of the multifractal spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we present a number of applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of inhomogeneous self-similar measures and again we present a number of applications of our results, in particular to non-linear self-similar measures.en
dc.format.extent1056016 bytes
dc.format.mimetypeapplication/pdf
dc.language.isoenen
dc.publisherUniversity of St Andrews
dc.rightsCreative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/
dc.subjectInhomogeneous self-similar setsen
dc.subjectInhomogeneous self-similar measuresen
dc.subjectHausdorff dimensionen
dc.subjectBox dimensionen
dc.subjectPacking dimensionen
dc.subjectL^q-spectraen
dc.subjectRenyi dimensionsen
dc.subjectMultifractal spectraen
dc.subjectFourier transformsen
dc.subject.lccQA248.S65
dc.subject.lcshSet theoryen_US
dc.subject.lcshHausdorff measuresen_US
dc.titleInhomogeneous self-similar sets and measuresen
dc.typeThesisen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD Doctor of Philosophyen
dc.publisher.institutionThe University of St Andrewsen


The following license files are associated with this item:

  • Creative Commons

This item appears in the following Collection(s)

Show simple item record

Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported
Except where otherwise noted within the work, this item's license for re-use is described as Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported