Projects per year
Abstract
We study Lq-spectra of planar self-affine measures generated by diagonal matrices. We introduce a new technique for constructing and understanding examples based on combinatorial estimates for the exponential growth of certain split binomial sums. Using this approach we disprove a theorem of Falconer and Miao from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for the generalised dimensions of generic self-affine measures. We also answer a question of Fraser from 2016 in the negative by proving that a certain natural closed form expression does not generally give the Lq-spectrum. As a further application we provide examples of self-affine measures whose Lq-spectra exhibit new types of phase transitions. Finally, we provide new nontrivial closed form bounds for the Lq-spectra, which in certain cases yield sharp results.
Original language | English |
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Pages (from-to) | 6331-6357 |
Number of pages | 27 |
Journal | Nonlinearity |
Volume | 34 |
Issue number | 9 |
Early online date | 2 Aug 2021 |
DOIs | |
Publication status | Published - Sept 2021 |
Keywords
- Fractals
- Lq-spectra
- Self-affine measures
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Dive into the research topics of 'Lq-spectra of self-affine measures: closed forms, counterexamples, and split binomial sums'. Together they form a unique fingerprint.Projects
- 2 Finished
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Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
1/02/18 → 11/06/21
Project: Standard
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Fractal Geometry and Dimension: Fractal Geometry and dimension theory
Fraser, J. (PI)
1/09/16 → 30/06/18
Project: Fellowship