The Dimension Spectrum of the Maximal Measure

A variety of complicated fractal objects and strange sets appears in nonlinear physics. In diffusion-limited aggregation, the probability of a random walker landing next to a given site of the aggregate is of interest. In percolation, the distribution of voltages across different elements in a random-resistor network (see [T. Halsey et al., Phys. Rev. A (3), 33 (1986), pp. 1141–1151]) may be of interest. These examples can be better analyzed by dividing certain objects in pieces labeled by indexes, but that leads to working with fractal sets and the notion of dimension [Halsey et al. (1986)]. The dimension spectrum of a system has been introduced and measured experimentally, and a substantial literature in physics addresses this topic. In several important cases, rigorous proofs of the results presented in [Halsey et al. (1986)] have been established. Here, rigorous mathematical proofs of some results in this theory are given, specifically for the maximal entropy measure of a hyperbolic rational map in the complex plane. In this case the fractal object is the Julia set (see [H. Brolin, Ark. Mat., 6 (1966), pp. 103–114], [A. Freire, A. Lopes, and R. Mañé, Bol. Soc.Brasil Mat., 14 (1983), pp. 45–62] ), which has been extensively studied in the physics literature.