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Generalized Farey Trees, Transfer Operators and Phase Transitions

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Abstract

We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral properties of the generalized transfer operator. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, I-40127, Bologna, Italy

    Mirko Degli Esposti

  2. Dipartimento di Matematica e Informatica, Università di Camerino, via Madonna delle Carceri, 62032, Camerino, Italy

    Stefano Isola

  3. Mathematisches Institut der Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, D-91054, Erlangen, Germany

    Andreas Knauf

Authors
  1. Mirko Degli Esposti
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  2. Stefano Isola
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  3. Andreas Knauf
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Correspondence to Andreas Knauf.

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Communicated by J.L. Lebowitz

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Degli Esposti, M., Isola, S. & Knauf, A. Generalized Farey Trees, Transfer Operators and Phase Transitions. Commun. Math. Phys. 275, 297–329 (2007). https://doi.org/10.1007/s00220-007-0294-3

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  • DOI: https://doi.org/10.1007/s00220-007-0294-3

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