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A Note on the Large Deviations for Piecewise Expanding Multidimensional Maps

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Nonlinear Dynamics New Directions

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 11))

Abstract

We present here the large deviation principle for some systems admitting a spectral gap, by using the functional approach of Hennion and Hervé, with slight modification. Our main application concerns multidimensional expanding maps introduced by Saussol.

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Notes

  1. 1.

    When \(\varphi \in{\cal B}^{\star}\) belongs to the topological dual of \({\cal B}\), we denote \(<\varphi, f> = \varphi(f)\). The linear form \(f \to \int f \, dm\) belongs to \({\cal B}^{\star}\), and we denote it by m.

  2. 2.

    See corollaries III.11 and III.6 in [17].

  3. 3.

    By checking the proof, we only need that \({\cal B}\) is a Banach algebra and \(\phi \in{\cal B}\) to prove that the operators P z are well defined and holomorphic in z. So we can just assume that ϕ is such that P z define a holomorphic family of bounded operators on \({\cal B}\) for z in a complex neighborhood of 0, with successive derivatives at 0 given by C n .

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Acknowledgement

S. Vaienti warmly thanks E. Ugalde for the kind invitation to participate in the conference in honor of Valentin Afraimovich; R. Aimino and S. Vaienti express their sincere gratitude to I. Melbourne who helped them to simplify the proofs. R. Aimino and S. Vaienti have been supported by the ANR-project Perturbations.

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Authors and Affiliations

  1. UMR-6207 Centre de Physique Théorique, CNRS, Universités d’Aix-Marseille, Université du Sud, Toulon-Var and FRUMAM, Fédéderation de Recherche des Unités de Mathématiques de Marseille CPT, Luminy Case 907, 13288, Marseille Cedex 9, France

    R. Aimino

  2. UMR-7332 Centre de Physique Théorique, CNRS, Universités d’Aix-Marseille, Université du Sud, Toulon-Var and FRUMAM, Fédéderation de Recherche des Unités de Mathématiques de Marseille CPT, Luminy Case 907, 13288, Marseille Cedex 9, France

    S. Vaienti

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  1. R. Aimino
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  1. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Hernán González-Aguilar

  2. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Edgardo Ugalde

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Aimino, R., Vaienti, S. (2015). A Note on the Large Deviations for Piecewise Expanding Multidimensional Maps. In: González-Aguilar, H., Ugalde, E. (eds) Nonlinear Dynamics New Directions. Nonlinear Systems and Complexity, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-09867-8_1

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  • DOI: https://doi.org/10.1007/978-3-319-09867-8_1

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