Second-order self-similar identities and multifractal decompositions (Q2710406)

In this paper net results are derived on the multifractal decomposition of self-similar measures on the line associated with a certain class of equicontractive iterated function systems with overlaps (that is, not fulfilling the open set condition). The results apply to the case \(\varphi_i(x)={1\over m}x+{m-1\over m}i\), \(i=0,1,2,\ldots,m,\) where \(m\) is an odd integer, which includes the setting of 3-fold convolutions of the Cantor measure (\(m=3\) and weights \(({1\over 8},{3\over 8},{3\over 8},{1\over 8})\)). NEWLINENEWLINENEWLINEClosed formulae for the \(L^q\)-spectra \(\tau(q)\) of the associated self-similar measures are obtained for \(q>0\), where \(\tau(q)\) are shown to be differentiable. Furthermore, the so-called multifractal formalism is justified to hold in the region \(q>0\). A key ingredient in the proofs are the ``second-order identities'' introduced in [\textit{R. S. Strichartz, A. Taylor} and \textit{T. Zhang}, Exp. Math. 4, No. 2, 101-128 (1995; Zbl 0860.28005)], first employed by the authors of the reviewed paper in deriving the multifractal structure of Bernoulli convolutions associated with the golden number [\textit{K.-S. Lau} and \textit{S.-M. Ngai}, Stud. Math. 131, No. 3, 225-251 (1998; Zbl 0929.28005)] and whose use is generalized here to study other overlapping self-similar measures.