Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.552198
Title: Inhomogeneous self-similar sets and measures
Author: Snigireva, Nina
Awarding Body: University of St Andrews
Current Institution: University of St Andrews
Date of Award: 2008
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Abstract:
The 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.
Supervisor: Olsen, Lars Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.552198  DOI: Not available
Keywords: Inhomogeneous self-similar sets ; Inhomogeneous self-similar measures ; Hausdorff dimension ; Box dimension ; Packing dimension ; L^q-spectra ; Renyi dimensions ; Multifractal spectra ; Fourier transforms ; QA248.S65 ; Set theory ; Hausdorff measures
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