Communications in Mathematical Physics

, Volume 300, Issue 1, pp 65–94 | Cite as

Natural Equilibrium States for Multimodal Maps



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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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Facultad de MatemáticasPontificia Universidad Católica de Chile (PUC)SantiagoChile
  2. 2.Departamento de Matemática PuraFaculdade de Ciências da Universidade do PortoPortoPortugal
  3. 3.Department of Mathematics and StatisticsBoston UniversityBostonUSA

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