Inhomogeneous self-similar sets and measures

Set theory;Hausdorff measures

Snigireva, Nina ; Olsen, Lars ; Olsen, Lars

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Set theory;    Hausdorff measures;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/682/Nina%20Snigireva%20PhD%20thesis.pdf?sequence=6&isAllowed=y

来源: DR-NTU

【 摘 要 】

The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets andmeasures. In particular, we show that these sets and measuresare natural generalizations of the well known self-similar sets andmeasures. We then investigate the structure of these sets and measures. In the second chapter we study various fractaldimensions (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-similarmeasures 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 mayhave phase transitions which is in sharp contrast to the behaviour of the L^{q} spectraof (ordinary) self-similarmeasures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectraof inhomogeneous self-similar measures. We show thatthe multifractal spectra ofinhomogeneous self-similarmeasuresmay be non-concave which is again in sharp contrast to the behaviour of the multifractal spectraof (ordinary) self-similarmeasures satisfying the Open Set Condition. Then we present a number ofapplications 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 ofinhomogeneous self-similar measures and again we present anumber of applications of our results, in particular to non-linearself-similar measures.