Dimension of Irregular Sets

Abstract

In Chapters 6 and 7 we described the main components of multifractal analysis for several multifractal spectra and several classes of dynamical systems. These spectra are obtained from multifractal decompositions such as the one in (7.1). In particular, we possess very detailed information from the ergodic, topological, and dimensional points of view about the level sets $K_\alpha ^g$ in each multifractal decomposition. On the other hand, we gave no nontrivial information about the irregular set in these decompositions, that is, the set X/Y in (7.1). Furthermore, the irregular set is typically very small from the point of view of ergodic theory. Namely, for many “natural” multifractal decompositions it has zero measure with respect to any finite invariant measure. Nevertheless, it may be very large from the topological and dimensional points of view. This is the main theme of this chapter, where we also describe a general approach to the study of the u-dimension of irregular sets.