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Sensitive Dependence of Geometric Gibbs States at Positive Temperature

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Abstract

We give the first example of a smooth family of real and complex maps having sensitive dependence of geometric Gibbs states at positive temperature. This family consists of quadratic-like maps that are non-uniformly hyperbolic in a strong sense. We show that for a dense set of maps in the family the geometric Gibbs states do not converge at positive temperature. These are the first examples of non-convergence at positive temperature in statistical mechanics or the thermodynamic formalism, and answers a question of van Enter and Ruszel. We also show that this phenomenon is robust: There is an open set of analytic 2-parameter families of quadratic-like maps that exhibit sensitive dependence of geometric Gibbs states at positive temperature.

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Acknowledgments

We would like to thank the referees for the comments and pointers to the literature. The first named author acknowledges partial support from FONDECYT Grant 1161221, and would like to thank the University of Rochester for its hospitality. The second named author acknowledges partial support from NSF Grant DMS-1700291, and would like to thank Universidad Andrés Bello for its hospitality.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Andres Bello, Avenida República 498, Santiago, Chile

    Daniel Coronel

  2. Department of Mathematics, University of Rochester, Hylan Building, Rochester, NY, 14627, USA

    Juan Rivera-Letelier

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  1. Daniel Coronel
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  2. Juan Rivera-Letelier
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Correspondence to Juan Rivera-Letelier.

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Communicated by C. Liverani

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Coronel, D., Rivera-Letelier, J. Sensitive Dependence of Geometric Gibbs States at Positive Temperature. Commun. Math. Phys. 368, 383–425 (2019). https://doi.org/10.1007/s00220-019-03350-6

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