Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and Banach space valued spectra

Abstract

During the past 10 years multifractal analysis has received an enormous interest. For a sequence ( φ n ) n of functions φ n : X → M on a metric space X, multifractal analysis refers to the study of the Hausdorff dimension of the level sets { x ∈ X | lim n φ n ( x ) = t } of the limit function lim n φ n . Previous studies have focused (almost) exclusively on the analysis of (1) multifractal spectra that depend linearly on the objects involved, of (2) so-called convergence points, i.e. points x for which the limit lim n φ n ( x ) exists, and, finally, of (3) functions φ n that take values in finite dimensional vector spaces. However, many important quantities describing the local structure of fractal measures and/or dynamical systems take values in infinite dimensional Banach spaces and/or depend in a highly non-linear way on the objects involved, and existing methods cannot be applied to the study of these quantities. Also, 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 ) involved do not exist; such points are called divergence points. In this paper we introduce and develope a general and unifying framework for (1) studying a very large and general class of non-linear multifractal spectra, for (2) providing a very detailed study of the fractal structure of individual divergence points, and, finally, for (3) studying multifractal spectra of general infinite dimensional Banach space valued functions. In particular, applications to new non-linear multifractal spectra of ergodic averages of Banach space valued functions are given. Also, applications to new multifractal spectra of ergodic averages and new multifractal spectra in metric number theory are presented.