Order parameter, symmetry breaking, and phase transitions in the description of multifractal sets

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Abstract
A new method is proposed to locate and analyze phase transitions in a thermodynamic formalism for the description of fractal sets. By studying order parameters appropriate to the transitions we get both an efficient numerical tool for locating phase transitions and an understanding of the structure of the ordered phase. With this method, we examine fractal sets generated by a class of maps of the interval close to the map x→4x(1-x). We show that the existence of phase transitions is a persistent phenomenon and remains when the map is perturbed although the structure of the entropy function changes drastically. For strong perturbations the transition disappears and the entropy function becomes nonsingular. The phase transitions describe transitions in the distribution of the characteristic Lyapunov exponents.