Multifractal analysis of infinite products
A multifractal formalism is formulated for a family of measures called infinite products, which generalize the Gibbs measures in the one-dimensional lattice gas model. If \(\{\lambda_n\}\) is a lacunary sequence of positive integers and \(\{g_n\}\) a sequence of positive periodic, continuous functions on the circle, then the infinite products are the weak limit points of the probability measures \(Z_n^{-1} P_n(t) dt\), where \(P_n(t)=\prod_{k=0}^{n-1} g_k(\lambda_k t)\) and \(Z_n\) is the normalizing factor. Under given conditions the infinite product is well defined and there is a family of Gibbs measures associated to it. The singularity spectrum formula describes the Hausdorff or packing dimension of the set of points of local dimension \(\alpha\) as the Legendre transform at \(\alpha\) of the so-called \(\tau\)-function. The main result of the paper shows that for infinite products under some conditions a weak form of the singularity spectrum formula holds. One of these conditions, the existence of the \(\varphi\)-function, is verified in important special cases, the others are explicit in \(\{\lambda_n\}\) and \(\{g_n\}\). It is also shown, that if the \(\tau\)-function is differentiable, the full singularity spectrum formula holds. These results imply new results on the multifractal analysis of Riesz products and the methods of the paper also extend to \(G\)-measures.
scientific article; zbMATH DE number 1279779