Entropy function for multifractals

Following an analogy to the formalism of statistical mechanics, an entropy function and a free energy are introduced for multifractals. These functions give a full description of the scaling behaviors of multifractals. The method of Halsey et al. [Phys. Rev. A 33, 1141 (1986)] for characterizing multifractals can naturally be interpreted by the use of these functions. For the invariant set of a dynamical system, these functions are furthermore related to the measure-theoretic (Kolmogolov-Sinai) entropy, the topological entropy, and the Lyapunov exponent.