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Exponential inequalities for dynamical measures of expanding maps of the interval
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  • Published: July 2002

Exponential inequalities for dynamical measures of expanding maps of the interval

  • P. Collet1,
  • S. Martinez2 &
  • B. Schmitt3 

Probability Theory and Related Fields volume 123, pages 301–322 (2002)Cite this article

Abstract.

 We prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval. As an immediate corollary we obtain a concentration inequality. We apply these results to the estimation of the rate of convergence of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure.

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Authors and Affiliations

  1. Centre de Physique Théorique, CNRS UMR 7644, Ecole Polytechnique, 91128 Palaiseau Cedex, France. e-mail: collet@cpht.polytechnique.fr, , , , , , FR

    P. Collet

  2. Universidad de Chile, Facultad de Ciencias Fí´sicas y Matemáticas, Departamento de Ingenierí´a Matemática and Centro de Modelamiento Matemático UMR 2071 UCHILE-CNRS, Casilla 170-3 Correo 3, Santiago, Chile. e-mail: smartine@dim.uchile.cl, , , , , , CL

    S. Martinez

  3. Université de Bourgogne, Département de Mathématiques, Faculté des Sciences Mirande, BP-138, 21004 Dijon Cedex, France. e-mail: schmittb@u-bourgogne.fr, , , , , , FR

    B. Schmitt

Authors
  1. P. Collet
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  2. S. Martinez
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  3. B. Schmitt
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Additional information

Received: 14 August 2001 / Revised version: 13 February 2002 / Published online: 1 July 2002

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Collet, P., Martinez, S. & Schmitt, B. Exponential inequalities for dynamical measures of expanding maps of the interval. Probab Theory Relat Fields 123, 301–322 (2002). https://doi.org/10.1007/s004400200204

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  • Issue Date: July 2002

  • DOI: https://doi.org/10.1007/s004400200204

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Keywords

  • Invariant Measure
  • Positive Measure
  • Dynamical Measure
  • Empirical Measure
  • Concentration Inequality
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