Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. III

Abstract

During the past 10 years multifractal analysis has received an enormous interest. For a sequence (φ n ) n of functions $$ \varphi _{n} :X \to M $$ on a metric space X, multifractal analysis refers to the study of the Hausdorff dimension of the level sets $$ {\left\{ {x \in X|{\mathop {\lim } \limits_n }\varphi _{n} (x) = t} \right\}} $$ of the limit function lim_n φ n . Previous studies have focused (almost) exclusively on the analysis of so-called convergence points, i.e. points x for which the limit lim_n φ n (x) exists. However, many important features describing the local structure of fractal measures and/or dynamical systems can be analyzed by investigating points x at which the limits lim_n φ n (x) do not exist; such points are called divergence points. In this paper we introduce and developed a general framework for performing a very detailed study of the fractal structure of individual divergence points. We define multifractal spectra that provides extremely precise quantitative information about the distribution of in- dividual divergence points thereby extending and unifying many diverse qualitative results on the behaviour of divergence points. Previous work have obtained information about the global structure of the set of divergence (namely the dimension of the set of all divergence points), whereas we obtain significantly finer and detailed information about the local fractal structure of the individual divergence points. In particular, applications to new multifractal spectra of ergodic averages and to new multifractal spectra in metric number theory are given.