The multifractal spectrum of quasi self-similar measures
A measure \(\mu\) on a compact metric space \((X,d)\) is said to be quasi self-similar provided that it is quasi self-similar from below as well as from above on the space \(X\). For example, \(\mu\) is quasi self-similar from below on \(X\) if there are \(0<R<1\) and \(m>2\) such that when \(0<r<R\) and \(x\in X\), there is a \(\phi: B(x,r)\to X\) satisfying that for all \(0<s\leq r\) and \(\zeta\in B(x,r)\) with \(B(\zeta,s)\subset B(x,r)\), \[ B(\phi\zeta, m^{-1}s/r)\subset\phi B(\zeta,s)\subset B(\phi\zeta, ms/r), \] and if \(A\subset B(x,r)\) is Borel then \[ m^{-1}{\mu(A)\over\mu B(x,r)}\leq \mu\phi(A)\leq m{\mu(A)\over\mu B(x,r)}. \] If \(\mu\) is quasi self-similar then it is shown that a coarse form, that is \[ \lim_{\sigma\searrow 0} \dim\{x\in\text{Spt } \mu:\alpha- \sigma\leq\underline\alpha_\mu(x)\leq\overline\alpha_\mu(x)\leq \alpha+\sigma\}= b^*(\alpha), \] of the multifractal formalism holds, where \(b^*\) denotes the Legendre transform of the generalized Hausdorff dimension function recently introduced by \textit{L. Olsen} [Adv. Math. 116, No. 1, 82-196 (1995; Zbl 0841.28012); ``Random geometrically graph directed self-similar multifractals'' (1994; Zbl 0801.28002)].
generalized Hausdorff and packing dimension
scientific article; zbMATH DE number 1046882
scientific article