Multifractal dimensions for invariant subsets of piecewise monotonic interval maps

Abstract

Abstract. The multifractal generalizations of Hausdorff dimension and packing dimension are investigated for an invariant subset A of a piecewise monotonic map on the interval. Formulae for the multifractal dimension of an ergodic invariant measure, the essential multifractal dimension of A, and the multifractal Hausdorff dimension of A are derived.

Dimension Theory for Multimodal Maps

“…Multifractal analysis of pointwise dimension was also considered in the countable Markov shift setting by Hanus et al [16] and Iommi [19]. For general piecewise continuous maps, analysis of this type was addressed in [18]. For multimodal maps, the multifractal analysis of pointwise dimension study began with the work of Todd [47].…”

Hausdorff and packing measure functions of self-similar sets: continuity and measurability

“…It is easily seen that the measure H q,t µ assigns a dimension to each subset E of R d in the usual way [Ol1]: there exists a unique number dim The number dim q H,µ (E) is an obvious multifractal analogue of the Hausdorff dimension dim H (E) of E. In fact, it follows immediately from the definitions that dim H (E) = dim 0 H,µ (E). The reader is referred to [Da,HRS,Ol2] and the references therein for a detailed discussion of the applications of these measures in multifractal analysis.…”

Empirical multifractal moment measures and moment scaling functions of self-similar multifractals

“…In an attempt to develop a general theoretical framework for studying the multifractal structure of Borel probability measures, [Ol1], [Pes1] and [Pey] introduced two families of measures, {H q,t µ | q, t ∈ R} and {P q,t µ | q, t ∈ R}, based on certain generalizations of the Hausdorff measure and of the packing measure. The measures H q,t µ and P q,t µ have subsequently been investigated further by a large number of authors, including [Co,Da1,Da2,Da3,HRS,Ol2,Ol3,O'N1,O'N2,Sche].…”