We present a quantitative theory of the scaling properties of Julia sets, using them as a case model for nontrivial fractal sets off the borderline of chaos. It is shown that generally the theory has a ‘‘macroscopic’’ part which consists of the generalized dimensions of the set, or its spectrum of scaling indexes, and a ‘‘microscopic’’ part which consists of scaling functions. These two facets are formally and computationally equivalent to thermodynamics and statistical mechanics in the theory of many-body systems. We construct scaling functions for the Julia sets and argue that basically there are two different approaches to this construction, which we term the Feigenbaum approach and the Ruelle-Bowen-Sinai approach. For the cases considered here the two approaches converge, meaning that we can map the theory onto Ising models with finite-range interactions. The largest eigenvalue of the appropriate transfer matrix furnishes the thermodynamic functions.

• Received 17 November 1986

DOI:https://doi.org/10.1103/PhysRevA.36.1409