2018/11/08 by Fraser, Jonathan M., Lee, Lawrence D., Morris, Ian D. +1
#15A18 #26A24 (secondary) #28A80 #37C45 (primary) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1811.03400
We study Lq-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 Lq-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 Lq-spectra exhibit new types of phase transitions. Our examples depend on a combinatorial estimate for the exponential growth of certain split binomial sums.