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Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle
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  • Published: June 1994

Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle

  • Zaqueu Coelho1 &
  • Pierre Collet2 

Probability Theory and Related Fields volume 99, pages 237–250 (1994)Cite this article

  • 86 Accesses

  • 18 Citations

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Summary

Given two pointsx, y∈S 1 randomly chosen independently by a mixing absolutely continuous invariant measure μ of a piecewise expanding and smooth mapf of the circle, we consider for each ε>0 the point process obtained by recording the timesn>0 such that |f n(x)−f n(y)|≦ε. With the further assumption that the density of μ is bounded away from zero, we show that when ε tends to zero the above point process scaled by ε−1 converges in law to a marked Poisson point process with constant parameter measure. This parameter measure is given explicity by an average on the rate of expansion off.

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Author information

Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 20570 (Ag. Iguatemi), 01498, São Paulo, SP, Brazil

    Zaqueu Coelho

  2. Ecole Polytechnique, Laboratoire UPR 14 du CNRS, Centre de Physique Théorique, F-91128, Palaiseau Cedex, France

    Pierre Collet

Authors
  1. Zaqueu Coelho
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  2. Pierre Collet
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Additional information

Partially supported by FAPESP grant number 90/3918-5

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Cite this article

Coelho, Z., Collet, P. Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle. Probab. Th. Rel. Fields 99, 237–250 (1994). https://doi.org/10.1007/BF01199024

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  • Received: 21 May 1993

  • Revised: 08 December 1993

  • Issue Date: June 1994

  • DOI: https://doi.org/10.1007/BF01199024

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Mathematics Subject Classification

  • 60F05
  • 60G55
  • 34C35
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