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arXiv · 1811.03400

$L^q$-spectra of self-affine measures: closed forms, counterexamples, and split binomial sums

Jonathan M. Fraser·Lawrence D. Lee·Ian D. Morris·Han Yu
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Abstract

We study $L^q$-spectra of planar self-affine measures generated by diagonal systems with an emphasis on providing closed form expressions. We answer a question posed by Fraser in 2016 in the negative by proving that a certain natural closed form expression does not generally give the $L^q$-spectrum and, using a similar approach, find counterexamples to a statement of Falconer-Miao from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for the generalised dimensions of generic self-affine measures. In the positive direction we provide new non-trivial closed form bounds in both of the above settings, which in certain cases yield sharp results. We also provide examples of self-affine measures whose $L^q$-spectra exhibit new types of phase transitions. Our examples depend on a combinatorial estimate for the exponential growth of certain split binomial sums.

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Jonathan M. Fraser, Lawrence D. Lee, Ian D. Morris, Han Yu. 2018-11-08. $L^q$-spectra of self-affine measures: closed forms, counterexamples, and split binomial sums. https://arxiv.org/abs/1811.03400

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