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Natural Equilibrium States for Multimodal Maps

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Abstract

This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials −t log |Df|, for the largest possible interval of parameters t. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained.

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Author notes

  1. Mike Todd

    Present address: Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA, 2215, USA

Authors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Avenida Vicuña Mackenna, 4860, Santiago, Chile

    Godofredo Iommi

  2. Departamento de Matemática Pura, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007, Porto, Portugal

    Mike Todd

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  1. Godofredo Iommi
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  2. Mike Todd
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Correspondence to Godofredo Iommi.

Additional information

Communicated by G. Gallavotti

GI was partially supported by Proyecto Fondecyt 11070050 and by Research Network on Low Dimensional Systems, PBCT/CONICYT, Chile. MT is supported by FCT grant SFRH/BPD/26521/2006 and also by FCT through CMUP.

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Iommi, G., Todd, M. Natural Equilibrium States for Multimodal Maps. Commun. Math. Phys. 300, 65–94 (2010). https://doi.org/10.1007/s00220-010-1112-x

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