The Thermodynamic Approach to Multifractal Analysis

Most results in multifractal analysis are obtained using either a thermodynamic approach based on existence and uniqueness of equilibrium states or a saturation approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of nonuniformly hyperbolic systems, including examples with phase transitions, and obtain new results for a number of examples that have already been studied using the saturation approach. We compare the thermodynamic and saturation approaches and give a survey of many of the multifractal results in the literature.