Abstract
We study L q -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 L q -spectrum. As a further application we provide examples of self-affine measures whose L q -spectra exhibit new types of phase transitions. Finally, we provide new non-trivial closed form bounds for the L q -spectra, which in certain cases yield sharp results.
Highlights
Introduction and summary of resultsThe Lq-spectrum is an important concept in multifractal analysis and quantifies global fluctuations in a given measure
We study Lq-spectra of planar self-affine measures generated by diagonal matrices
As a further application we provide examples of self-affine measures whose Lq-spectra exhibit new types of phase transitions
Summary
The Lq-spectrum is an important concept in multifractal analysis and quantifies global fluctuations in a given measure. The paper [3] was mainly concerned with dimensions of self-affine sets, but towards the end it states a closed form expression for the generalised q-dimensions (these are a normalised version of the Lq-spectra) in a natural generic setting [3, theorem 4.1]. The proof of this result was just sketched and when the result appeared later in Miao’s thesis [8, theorem 3.11] the full proof was only given for 0 < q < 1 and the formula only conjectured to hold for q > 1. A key technical tool is the following growth result for split binomial sums: if one considers the binomial expansion of (1 + x)k, where x > 1 is fixed, and splits the sum in half, the ratio of the two halves grows exponentially in k, see theorem 2.1
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