The thermodynamic approach to multifractal analysis (Q2928244)

A thermodynamic approach and an orbit-gluing approach are often used in multifractal analysis to obtain a variety of results. The first one is based on the existence and uniqueness of equilibrium states and the second one is based on some version of the specification property. In 2001, Barreira and Saussol introduced a general way of taking advantage of the most important multifractal spectra. In particular, they used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting. The author of the present survey has two goals. First, he considers multifractals as described by Barreira, Pesin and Schmeling and generalizes a number of results from Barreira's and Saussol's work. More precisely, he offers purely thermodynamic conditions for the multifractal formalism to hold by studying the topological pressure. Consequently, multifractal analysis is extended to a wide-class of non-uniformly hyperbolic systems including examples with phase transitions. He also uses the orbit-gluing approach to prove new results for spectra and systems already known in the literature. Second, he examines the relationship between the thermodynamic approach to multifractals and the orbit-gluing approach by explaining the similarities and distinctions in various results in the literature.