$L^q$-spectra of self-affine measures: closed forms, counterexamples, and split binomial sums
Abstract
We study -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 -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 -spectra exhibit new types of phase transitions. Our examples depend on a combinatorial estimate for the exponential growth of certain split binomial sums.
Cite
@article{arxiv.1811.03400,
title = {$L^q$-spectra of self-affine measures: closed forms, counterexamples, and split binomial sums},
author = {Jonathan M. Fraser and Lawrence D. Lee and Ian D. Morris and Han Yu},
journal= {arXiv preprint arXiv:1811.03400},
year = {2018}
}
Comments
27 Pages, 4 Figures