Multifractal analysis and phase transitions for hyperbolic and parabolic horseshoes
The paper presents a complete study of the thermodynamic formalism, the entropy spectrum of Birkhoff averages, and the ergodic optimization problem for a family of parabolic horseshoes. Roughly speaking, parabolic horseshoes are invariant sets topologically equivalent to hyperbolic horseshoes, although they lack hyperbolicity (or at least uniform hyperbolicity) at one or more points. The paper considers the model case of parabolic horseshoes for which the hyperbolicity breaks down at a single point. It should be emphasized that even in this situation the results are substantially different from those in the case of hyperbolic horseshoes. The paper also considers a large class of potentials that are not necessarily regular. For nonregular potentials in this class both the uniqueness of equilibrium measures and the occurrence of phase transitions are described. The approach consists in reducing the problems to the study of renewal shifts. Applications of this approach to hyperbolic horseshoes and to noninvertible maps, both parabolic and uniformly expanding, are presented. This allows to recover in a unified manner several results available in the literature. For the family of hyperbolic horseshoes, the dimension spectrum of equilibrium measures of a class of potentials that are not necessary regular is described. In particular, it is shown that the dimension spectra need not be strictly convex.
scientific article; zbMATH DE number 5884606
scientific article