Abstract
We effect a complete study of the thermodynamic formalism, the entropy spectrum of Birkhoff averages, and the ergodic optimization problem for a family of parabolic horseshoes. We consider a large class of potentials that are not necessarily regular, and we describe both the uniqueness of equilibrium measures and the occurrence of phase transitions for nonregular potentials in this class. Our approach consists in reducing the problems to the study of renewal shifts. We also describe applications of this approach to hyperbolic horseshoes as well as to noninvertible maps, both parabolic (with the Manneville-Pomeau map) and uniformly expanding. This allows us to recover in a unified manner several results scattered in the literature. For the family of hyperbolic horseshoes, we also describe the dimension spectrum of equilibrium measures of a class of potentials that are not necessarily regular. In particular, the dimension spectra need not be strictly convex.
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Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundaçcão para a Ciência e a Tecnologia by Program POCTI/FEDER, and the grant SFRH/BPD/21927/2005. G.I. was also supported by Proyecto Fondecyt 11070050 and by Research Network on Low Dimensional Dynamics, CONICYT, Chile.
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Barreira, L., Iommi, G. Multifractal analysis and phase transitions for hyperbolic and parabolic horseshoes. Isr. J. Math. 181, 347–379 (2011). https://doi.org/10.1007/s11856-011-0013-9
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DOI: https://doi.org/10.1007/s11856-011-0013-9