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Equilibrium State for One-Dimensional Lorenz-Like Expanding Maps

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Abstract

Let \(L:[0,1]{\setminus }\{d\}\rightarrow [0,1]\) be a one-dimensional Lorenz-like expanding map (d is the point of discontinuity), \(\mathcal {P}=\{ (0,d),(d,1) \}\) and \(C^{\alpha }([0,1],{\mathcal {P}})\) the set of piecewise Hölder-continuous potentials of [0, 1] with the usual \(\mathcal {C}^0\) topology. In this context, applying a criteria by Buzzi and Sarig (Ergod Theory Dyn Syst 23(5):1383–1400, 2003, Th. 1.3), we prove that there exists an open and dense subset \(\mathcal {H}\) of \(C^{\alpha }([0,1],{\mathcal {P}})\), such that each \(\phi \in \mathcal {H}\) admits exactly one equilibrium state.

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Acknowledgements

We thank Ali Tahzibi for proposing the problem and for many helpful suggestions during the preparation of the paper. Also, we thank Daniel Smania and Krerley Oliveira for some helpful conversations and comments on the problem.

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  1. Faculdade de Matemática, FAMAT-UFU, Uberlândia, MG, Brazil

    M. A. Bronzi & J. G. Oler

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  1. M. A. Bronzi
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  2. J. G. Oler
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Correspondence to M. A. Bronzi.

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Bronzi, M.A., Oler, J.G. Equilibrium State for One-Dimensional Lorenz-Like Expanding Maps. Bull Braz Math Soc, New Series 49, 873–892 (2018). https://doi.org/10.1007/s00574-018-0084-x

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