Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension

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 and/or packing dimension of the level sets (1) { x ∈ X | lim n φ n ( x ) = t } of the limit function lim n φ n . However, recently a more general notion of multifractal analysis, focusing not only on points x for which the limit lim n φ n ( x ) exists, has emerged and attracted considerable interest. Namely, for a sequence ( x n ) n in a metric space X, we let A ( x n ) denote the set of accumulation points of the sequence ( x n ) n . The problem of computing that the Hausdorff dimension of the set of points x for which the set of accumulation points of the sequence ( φ n ( x ) ) n equals a given set C, i.e. computing the Hausdorff dimension of the set (2) { x ∈ X | A ( φ n ( x ) ) = C } has recently attracted considerable interest and a number of interesting results have been obtained. However, almost nothing is known about the packing dimension of sets of this type except for a few special cases investigated in [I.S. Baek, L. Olsen, N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math. 214 (2007) 267–287]. The purpose of this paper is to compute the packing dimension of those sets for a very general class of maps φ n , including many examples that have been studied previously, cf. Theorem 3.1 and Corollary 3.2. Surprisingly, in many cases, the packing dimension and the Hausdorff dimension of the sets in (2) do not coincide. This is in sharp contrast to well-known results in multifractal analysis saying that the Hausdorff and packing dimensions of the sets in (1) coincide.