The mixed \(L^{q}\)-spectra of self-conformal measures satisfying the weak separation condition
The mixed \(L^q\)-spectra of measures \(\mu_1,\dots, \mu_k\) is denoted by \(D(q)=\overline D(q)=\underline D(q)\), with the upper \(\overline D(q)= \limsup_{r\to 0}{\log I(r,q)\over -\log r}\) and lower \(\underline D(q)= \liminf_{r\to 0} {\log I(r,q)\over -\log r}\) mixed \(L^q\)-spectra, and \[ I(r,q)= \int \mu_1(B(x_1, r))^{q_1}\cdots \mu_k(B(x_k, r))^{q_k} d\mu(x_1,\dots, x_k). \] Theorem. Let \((\{S_i\}^N_{i=1},\{p_{ji}\}^N_{i=1})\), \(j= 1,\dots, k\), be conformal iterated function system that have the bounded distortion property and satisfy the weak separation condition), then \(\overline D(q)=\underline D(q)\equiv\beta(q)\), \(q= (q_1,\dots, q_k)\in \mathbb{R}^k\).
scientific article; zbMATH DE number 5925545
scientific article