Bounds for the \(L^q\)-spectra of a self-similar multifractal not satisfying the open set condition
Let \(S_i:\mathbb R^d\to \mathbb R^d\) be contracting similarities with contraction ratios \(r_i\) for \(i=1,\dots, N\), and let \((p_1, \dots, p_N)\) be a probability vector. Denote by \(K\) and \(\mu\) be the self-similar set and the self-similar measure associated with the list \((S_1,\dots, S_N, p_1, \dots, p_N)\). For any \(q\in \mathbb R\), define the function \(\beta: \mathbb R\to \mathbb R\) by \[ \sum_{i\in I} p_i^qr_i^{\beta(q)}=1 \] and define the lower and upper \(L^q\)-spectra of \(\mu\) as \[ \underline{\tau}_{\mu}(q)= \liminf_{r\to 0} \frac{\log I_{\mu}(r,q)}{-\log r}, \quad \overline{\tau}_{\mu}(q)=\limsup_{r\to 0}\frac{\log I_{\mu}(r,q)}{-\log r}, \] where \(I_{\mu}(r,q)= \int_K\mu (B(x,r))^{q-1}\,d\mu(x)\) and \(B(x,r)\) denotes the open ball with centre \(x\) and radius \(r\). The paper under review investigates \(L^q\)-spectra of a self-similar measure without assuming the open set condition. Non-trivial bounds for the \(L^q\)-spectra of an arbitrary self-similar measure are obtained. Theorem. For a positive integer \(n\), let \[ \mathcal {I}_n=\Big\{I\subseteq \Sigma^n\;\Big|\;\bigcap_{{\mathbf{i}}\in I}K_{\mathbf{i}} \neq \varphi\Big\}. \] There exists a unique \(s_n\in \mathbb{R}\) such that \[ 1=\max_{I\in \mathcal {I}_n} \sum_{{\mathbf i}\in I} p_{\mathbf i}r_{\mathbf{i}}^{-s_n}. \] Let \(s=\sup_ns_n\). For all \(q\in \mathbb{R}\), with \(q\geq 1\), we have \[ \beta(q)\leq \underline{\tau}_{\mu}(q)\leq \overline{\tau}_{\mu}(q)\leq s(1-q). \] At the same time, an analogous result for mixed \(L^q\)-spectra is given.
scientific article; zbMATH DE number 5555231
scientific article