The exact rate of convergence of the Lq-spectra of self-similar measures for q<0

The Lq-spectrum of a Borel measure is one of the key objects in multifractal analysis, and it is widely believed that Lq-spectrum associated with a fractal measure encode important information about the underlying dynamics and geometry. The study of the Lq-spectrum therefore plays a fundamental role in the understanding of dynamical systems or fractal measures. For q⩾0 Olsen [L. Olsen, Empirical multifractal moment measures and moment scaling functions of self-similar multifractals, Math. Proc. Cambridge Philos. Soc. 133 (2002) 459–485] recently determined the exact rate of convergence of the Lq-spectra of a self-similar measure satisfying the Open Set Condition (OSC). Unfortunately, nothing is known about the rate of convergence for q<0. Indeed, the problem of analysing Lq-spectra for q<0 is generally considered significantly more difficult since the Lq-spectra are extremely sensitive to small variations in the distribution of μ for q<0. The purpose of this paper is to overcome these obstacles and to investigate the more difficult problem of determining the exact rate of convergence of the multifractal Lq-spectra of a self-similar measure satisfying the OSC for q<0.