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Equilibrium States of Weakly Hyperbolic One-Dimensional Maps for Hölder Potentials

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Abstract

There is a wealth of results in the literature on the thermodynamic formalism for potentials that are, in some sense, “hyperbolic”. We show that for a sufficiently regular one-dimensional map satisfying a weak hyperbolicity assumption, every Hölder continuous potential is hyperbolic. A sample consequence is the absence of phase transitions: The pressure function is real analytic on the space of Hölder continuous functions. Another consequence is that every Hölder continuous potential has a unique equilibrium state, and that this measure has exponential decay of correlations.

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Author notes

  1. Huaibin Li

    Present address: School of Mathematics and Information Science, Henan University, 475004, Kaifeng, China

Authors and Affiliations

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

    Huaibin Li & Juan Rivera-Letelier

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  1. Huaibin Li
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  2. Juan Rivera-Letelier
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Correspondence to Juan Rivera-Letelier.

Additional information

Communicated by M. Lyubich

HL was partially supported by the National Natural Science Foundation of China (Grant No. 11101124) and FONDECYT Grant 3110060 of Chile.

JRL was partially supported by FONDECYT Grant 1100922 of Chile.

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Li, H., Rivera-Letelier, J. Equilibrium States of Weakly Hyperbolic One-Dimensional Maps for Hölder Potentials. Commun. Math. Phys. 328, 397–419 (2014). https://doi.org/10.1007/s00220-014-1952-x

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