Skip to main content
SpringerLink
Log in

Limit laws of entrance times for homeomorphisms of the circle

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Given a homeomorphismf of the circle with irrational rotation number and a descending chain of renormalization intervalsj n off, we consider for each interval the point process obtained by marking the times for the orbit of a point in the circle to enterJ n. Assuming the point is randomly chosen by the unique invariant probability measure off, we obtain necessary and sufficient conditions which guarantee convergence in law of the corresponding point process and we describe all the limiting processes. These conditions are given in terms of the convergent subsequences of the orbit of the rotation number off under the Gauss transformation and under a certain realization of its natural extension. We also consider the case when the point is randomly chosen according to Lebesgue measure,f being a diffeomorphism which isC 1-conjugate to a rotation, and we show that the same necessary and sufficient conditions guarantee convergence in this case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • [Br] L. Breiman,Probability theory, Addison-Wesley Series in Statistics, 1968.

  • [CC] Z. Coelho and P. Collet,Limit law for the close approach of two trajectories of expanding maps of the circle, Probability Theory and Related Fields99 (1994), 237–250.

    Article  MATH  MathSciNet  Google Scholar 

  • [CG] P. Collet and A. Galves,Asymptotic distribution of entrance times for expanding maps of an interval, preprint (1992).

  • [DV] D. J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes, Springer Series in Statistics, Springer-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  • [dF] E. de Faria,Proof of universality for critical circle mappings, PhD Thesis, CUNY (1992).

  • [He] M. Herman,Sur la conjugaison differentiable des difféomorphismes du cercle a des rotations, Publications Mathématiques de l’IHES49 (1979), 5–234.

    MATH  MathSciNet  Google Scholar 

  • [Hi] M. Hirata,Poisson law for Axiom A diffeomorphisms, Ergodic Theory and Dynamical Systems13 (1993), 533–556.

    MATH  MathSciNet  Google Scholar 

  • [IN] Sh. Ito and H. Nakada,On natural extensions of transformations related to Diophantine approximations, inNumber Theory and Combinatorics (J. Akiyamaet al., eds.), World Scientific, Singapore, 1985, pp. 185–205.

    Google Scholar 

  • [La] O.E. Lanford,Renormalization group methods for circle mappings, inNonlinear Evolution and Chaotic Phenomena, NATO Adv. Sci. Inst. Ser. B: Phys.,176, Plenum, New York, 1988, pp. 25–36.

    Google Scholar 

  • [MP] R.S. MacKay and I.C. Percival,Universal small-scale structure near the boundary of Siegel disks of arbitrary rotation number, Physica26D (1987), 193–202.

    MathSciNet  Google Scholar 

  • [Ne] J. Neveu,Processus pontuels, Springer Lecture Notes in Mathematics598 (1976), 249–445.

    Article  MathSciNet  Google Scholar 

  • [Pi] B. Pitskel,Poisson limit law for Markov chains, Ergodic Theory and Dynamical Systems11 (1991), 501–513.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 20570, 01452-990, São Paulo, SP, Brazil

    Zaqueu Coelho & Edson de Faria

Authors
  1. Zaqueu Coelho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Edson de Faria
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zaqueu Coelho.

Additional information

This work is part of Projeto Temático de Equipe “Transição de Fase Dinâmica em Sistemas Evolutivos” financially supported by FAPESP grant 90/3918-5.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Coelho, Z., de Faria, E. Limit laws of entrance times for homeomorphisms of the circle. Israel J. Math. 93, 93–112 (1996). https://doi.org/10.1007/BF02761095

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02761095

Keywords

Search

Navigation