Natural equilibrium states for multimodal maps

Details Natural equilibrium states for multimodal maps Natural equilibrium states for multimodal maps

2009-07-14

arxiv.org/abs/0907.2406v1

Communications in Mathematical Physics 300, 65-94 (2010)

Mathematics

Dynamical Systems

Scientific paper

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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