The correlation dimension of the Sierpinski carpet
Multifractal analysis is concerned with the decay rates of the measures \(\mu(B(x,r))\) of balls of radius \(r\) as \(r\) goes to 0 in a metric space \(X\), where \(f:X \to X\) is surjective, and \(\mu\) is an \(f\)-invariant measure on \(X\). The correlation dimension of the measure \(\mu\) is defined for each real \(\beta\) by \[ C(\beta) = \lim_{r \to 0} {Log \int \mu(B(x,r))^ \beta \mu (dx)\over Log r}. \] The correlation dimension is in general different from other characteristic dimensions such as Hausdorff dimension, capacity, or information dimension. Using a standard thermodynamic formalism, the author introduces partition functions (associated with partitions \((Q_ k)_{k \geq 1}\) whose diameters tend to 0) which are defined for real \(\beta\) by \(Z_ k(\beta) = \sum_{u \in Q_ k} \mu(u)^ \beta\) and a free-energy function \[ F(\beta) = -\lim_{k \to \infty} {1\over k} Log_ m Z_ k(\beta). \] The author proves that \(C(\beta) = F(\beta + 1)\) for all real \(\beta\) for Sierpinski carpets. Measures of Gibbs type, as well as generalizations to Markovian measures are considered.
scientific article; zbMATH DE number 716239
scientific article