Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages

We introduce and develope a unifying multifractal framework. The framework developed in this paper is based on the notion of deformations of empirical measures. This approach leads to significant extensions of already know results. However, our approach not only leads to extensions of already know results, but also, by considering non-linear deformations, provides the basis for the study of several new and non-linear local characteristic. We also initiate a detailed study of the fractal structure of so-called divergence points. We define multifractal spectra that provides extremely precise quantitative information about the distribution of individual divergence points of arbitrary (possibly non-linear) deformations, thereby extending and unifying many diverse qualitative results on the behaviour of divergence points. The techniques used in proving the main results are taken from large deviation theory and are completely different from previous techniques in the literature. (C) 2003 Elsevier SAS. All rights reserved.