A Multifractal Formalism for Hewitt–Stromberg Measures

Abstract

In the present work, we give a new multifractal formalism for which the classical multifractal formalism does not hold. We precisely introduce and study a multifractal formalism based on the Hewitt-Stromberg measures and that this formalism is completely parallel to Olsen's multifractal formalism which based on the Hausdorff and packing measures.The multifractal formalism (1.1) has been proved rigorously for random and non-random self-similar measures [1,16,42,43, 51], for self-conformal measures [26,27,28,29,37,52], for self-affine measures [6,7,8,9,23,24,36,45] and for Moran measures [61,62,63,64]. We note that the proofs of the multifractal formalism (1.1) in the above-mentioned references [1,10,12,13,14,36,37,42,43,45,52] are all based on the same key idea. The upper bound for f µ (α) is obtained by a standard covering argument (involving Besicovitch's Covering Theorem 2000 Mathematics Subject Classification. 28A78, 28A80.

On the projections of the multifractal packing dimension for $$q>1$$

“…The notion of singularity exponents or spectrum and generalized dimensions are the major components of the multifractal analysis. They were introduced with a view of characterizing the geometry of measure and to be linked with the multifractal spectrum which is the map which affects the Hausdorff or packing dimension of the iso-Hölder sets E µ (α) = x ∈ supp µ lim r→0 log µB(x, r) log r = α for a given α ≥ 0 and supp µ is the topological support of probability measure µ on R n , B(x, r) is the closed ball of center x and radius r. It unifies the multifractal spectra to the multifractal packing function b µ (q) and the multifractal packing function B µ (q) via the Legendre transform [1,3,8,38], i.e.,…”

“…Such measures appear also explicitly, for example, in Pesin's monograph [42, 5.3] and implicitly in Mattila's text [36]. Motivated by the above papers, the authors in [2,3] introduced and studied a multifractal formalism based on the Hewitt-Stromberg measures. However, we point out that this formalism is completely parallel to Olsen's multifractal formalism introduced in [38,39] which is based on the Hausdorff and packing measures.…”

“…Section 2.1 recalls the multifractal formalism introduced in [41]. In Section 2.2 we introduce the multifractal Hewitt-Stromberg measures and separator functions which slightly differ from those introduced in [2,3], and study their properties. Section 3 contain our main results.…”

“…The measures H q,t µ and P q,t µ and the pre-measures L q,t µ and C q,t µ assign in the usual way a multifractal dimension to each subset E of R n , they are respectively denoted [3,32,40] for precise definitions of these dimensions.…”

“…the multifractal box dimensions. The dimension functions b µ and B µ are intimately related to the spectra functions f µ and F µ (see [3]), whereas the dimension function ∆ µ is closely related to the upper box spectrum (more precisely, to the upper multifractal box dimension function τ µ , see Proposition 5.2).…”