An improved multifractal formalism and self-similar measures
Suppose that a partition of \(\mathbb{R}^ d\) into boxes \(B\) of side \(\delta\) is given. For a Borel measure \(\mu\) on \(\mathbb{R}^ d\) and a real number \(q\) the generalized dimension \(d_ q\) is defined by \[ \begin{aligned} s_ \delta(q) &= \sum_{\mu (B)\neq 0} \mu(b)^ q,\\ \tau(q) &= \limsup_{\delta\downarrow 0} {{\log s_ \delta(q)} \over {-\log \delta}} \quad (\text{singularity exponent}),\\ d_ q &= {{\tau(q)} \over {1-q}} \quad (q\neq 1). \end{aligned} \] The author shows by an example that this definition is unsatisfactory since it gives in some cases \(\tau(q)= \infty\) for all \(q<0\). The improved formalism of the author replaces only the boxes of side \(\delta\) by boxes \((B)_ \kappa\) of side \((2\kappa +1)\delta\) (later \(\kappa=1\) is fixed). The corresponding new notions to those above are \(S_ \delta\), \(T\) and \(D_ q\). It turns out that for \(q\geq 0\) nothing is changed. Later on, the author modifies the usual spectrum definition as well and establishes the known relation between new spectrum and new singularity exponent via Legendre transformation. The consequences of this simple modification for self- similar measures are discussed in detail, for example if \(\mu\) is a self- similar measure with ratios \(\lambda_ 1, \dots, \lambda_ r\) and probabilities \(p_ 1,\dots, p_ r\) then the index \(T(q)\) satisfies now for all \(q\in \mathbb{R}\) \[ \sum_{i=1}^ r p^ q_ i \lambda_ i^{T(q)} =1. \]
scientific article; zbMATH DE number 743912