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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[Br] Breiman, L.: Probabolity theory. Addison-Wesley 1968
[Co] Collet, P. Some ergodic properties of maps of the interval. In: Bamon R. et al.: Dynamical Systems and Frustrated Systems. (to appear)
[CG] Collet, P. and Galves, A.: Asymptotic distribution of entrance times for expanding maps of an interval. (Preprint 1992)
[DV] Daley, D.J. and Vere-Jones, D.: An introduction to the theory of point processes (Ser. Stat.) Berlin Heidelberg New York: Springer 1988
[Do] Donoghue, W.F., Jr.: Monotone matrix functions and analytic continuation (Grundlehren Math. Wiss. B207) Berlin Heidelberg New York: Springer 1974
[Hi] Hirata, M.: Poisson law for Axiom A diffeomorphisms. University of Tokyo (Preprint 1991)
[HK] Hofbauer, F., and Keller, G.: Ergodic properties of piecewise monotonic transformations, Math. Z.180, 119–140 (1982)
[LY] Lasota, A., and Yorke, J.A.: On the existence of invariant measures for piecewise monotonic transformations. Trans. Am. Math. Soc.186, 481–488 (1973)
[Ne] Neveu, J.: Processus pontuels. (Lect. Notes in Math., vol.598, pp. 249–445) Berlin Heidelberg New York: 1976
[Pi] Pitskel, B.: Poisson limit law for Markov chains. Ergodic Theory and Dyn. Syst.11, 501–513 (1991)
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Partially supported by FAPESP grant number 90/3918-5
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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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DOI: https://doi.org/10.1007/BF01199024