The singularity spectrum of the inverse of cookie-cutters

Abstract

Abstract. Gibbs measures µ on cookie-cutter sets are the archetype of multifractal measures on Cantor sets. In this article we compute the singularity spectrum of the inverse measure of µ. Such a measure is discrete (it is constituted only by Dirac masses), it satisfies a multifractal formalism, and its L q -spectrum possesses one point of non differentiability. The results rely on heterogeneous ubiquity theorems.

Multifractal analysis for projections of Gibbs and related measures

“…The Legendre transform of τ µ is defined as τ * µ : α ≥ 0 → inf q∈R αq − τ µ (q). The multifractal formalism also holds fully for classes of discrete measures whose weights are built by using measures in the previous classes [4,5,7,19,36]. This formalism turns out to hold on the whole interval τ * µ −1 (R + ), for some classes of Gibbs (or weak Gibbs) like measures [8,13,22,28,45,[52][53][54][55] and statistically self-conformal measures [1,3,9,14,16,18,32,44,45,50].…”

A Multifractal Formalism for Hewitt–Stromberg Measures

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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.

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Multifractal formalism for the inverse of random weak Gibbs measures

“…(2) Although the main lines of the proof of theorem 2.1(i) are similar to those used to treat the case of deterministic Gibbs measures [7], the study of ν ω requires the tools developed in [22] to study the multifractal nature of random weak Gibbs measures. Also, it is made structurally more complex because the weak Gibbs measures are constructed on a random subshift; this is reflected in the expression of ν ω as a weighted sum of Dirac masses (see propositions 4 and 5).…”

“…With this definition, they observe that for a Gibbs measure on a cookie-cutter set, while it is well known that the strong multifractal formalism holds, they can establish its failure on a non trivial interval for the discrete inverse measure of such a measure (they obtained the same type of failure for discrete in-homogeneous self-similar measures, see also [20]). Later, the validity of the multifractal formalism as defined above was obtained in [7], where the authors used the so-called heterogeneous, or conditioned, ubiquity theory, which combines ergodic theory and metric approximation theory, and was developed in [4]. This tool makes it possible to study a broad class of multifractal discrete measures [2,6].…”

“…The precise definitions of these objects and our main result, theorem 2.1, are exposed in the next section. Let us just mention that the randomness and the fact that we work on a subshift rather than a fullshift are two sources of serious complications with respect to the study achieved for inverse measures of deterministic Gibbs measures in [7]. In particular, the fundamental geometric tool provided by [4] must be revisited.…”