L^q-SPECTRUM OF A CLASS OF SELF-SIMILAR MEASURES^∗

We compute the L^q-spectrum of self-similar measures defined by an iterated function system of the form S_i (x) = (x + i)/2, i = 0, 1, …, m, m ≥ 2. For an iterated function system of the form S_i (x) = (x + (N − 1)i)/N, i = 0, 1, …, N, N ≥ 3, the L^q-spectrum of a corresponding self-similar measure was computed by Lau and Ngai [Indiana Univ. Math. J. 49 (2000), 925–972] for q ≥ 0 and by Feng, Lau and Wang [Asian J. Math. 9 (2005), 473–488] for the case N = 3 and q < 0. The method used by Lau and Ngai fails if the contraction ratio is 1/2. We use some techniques by Feng, Lau and Wang to express the L^q-spectrum of µ as a limit of matrix products. By defining a sub-multiplicative sequence and using its properties, we obtain a formula for the L^q-spectrum, q ∈ R, under the assumption that some limit function r(q) exists when q < 0. We study systems for which the limit defining r(q) exists.