Analytic evaluation of the multifractal properties of a Newtonian Julia set

The Julia set associated with the Newtonian map for the solution of $z^3$-1=0 in the complex plane is investigated by purely analytic means. The hierarchical organization of this strange repeller is explained from basic principles. The family of generalized fractal dimensions and the equivalent spectrum of scaling indices is evaluated using an extension of the Hentschel-Procaccia renormalization scheme. Our quantitative results are in excellent agreement with direct numerical computations.