Skip to main content
SpringerLink
Log in

Regular or stochastic dynamics in real analytic families of unimodal maps

  • Published:
Inventiones mathematicae Aims and scope

Abstract

In this paper we prove that in any non-trivial real analytic family of quasiquadratic maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, “hybrid classes”. This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family.

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

  1. Ahlfors, L.: Lectures on quasi-conformal maps. Van Nostrand Co 1966

  2. Ahlfors, L., Bers, L.: Riemann mapping theorem for variable metrics. Ann. Math. 72, 385–404 (1960)

    MATH  Google Scholar 

  3. Avila, A.: Bifurcations of unimodal maps: the topologic and metric picture. IMPA Thesis (2001) (http://www.math.sunysb.edu/∼artur/)

  4. Avila, A., Moreira, C.G.: Statistical properties of unimodal maps: the quadratic family. Preprint (http://www.arXiv.org). To appear in Ann. Math.

  5. Avila, A., Moreira, C.G.: Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative. Preprint (http://www.arXiv.org). To appear in Astérisque

  6. Avila, A., Moreira, C.G.: Statistical properties of unimodal maps: periodic orbits, physical measures and pathological laminations. Preprint (http://www.arXiv.org)

  7. Benedicks, M., Carleson, L.: On iterations of 1-ax 2 on (-1,1). Ann. Math. 122, 1–25 (1985)

    MathSciNet  MATH  Google Scholar 

  8. Bers, L.: The moduli of Kleinian groups. Russ. Math. Surv. 29, 86–102 (1974)

    MATH  Google Scholar 

  9. Bers, L., Royden, H.L.: Holomorphic families of injections. Acta Math. 157, 259–286 (1986)

    MathSciNet  MATH  Google Scholar 

  10. Branner, B., Hubbard, J.H.: The iteration of cubic polynomials. II: Patterns and parapatterns. Acta Math. 169, 229–325 (1992)

    MATH  Google Scholar 

  11. Collet, P., Eckmann, J.-P.: Positive Liapunov exponents and absolute continuity for maps of the interval. Ergodic Theory Dyn. Syst. 3, 13–46 (1983)

    MathSciNet  MATH  Google Scholar 

  12. Douady, A.: Prolongement de mouvements holomorphes (d’après Slodkowski et autres). Astérisque 227, 7–20 (1995)

    MATH  Google Scholar 

  13. Douady, A.: Systèmes dynamiques holomorphes. Bourbaki seminar, Vol. 1982/83, 39–63, Astérisque 105–106 (1983)

  14. Douady, A., Hubbard, J.H.: On the dynamics of polynomial-like maps. Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 287–343 (1985)

    Google Scholar 

  15. Douady, A., Hubbard, J.H.: Étude dynamique des polynômes complexes. Parties I et II. Publ. Math. Orsay 84–2 (1984) & 85–4 (1985)

  16. Earle, C., McMullen, C.: Quasiconformal isotopies. In: Holomorphic Functions and Moduli I. Springer MSRI publications volume 10, 143–154 (1988)

    MATH  Google Scholar 

  17. Fatou, P.: Sur les équations fonctionnelles. Bull. Soc. Math. Fr. 47 (1919)

  18. Guckenheimer, J.: Sensitive dependence to initial conditions for one-dimensional maps. Commun. Math. Phys. 70, 133–160 (1979)

    MathSciNet  MATH  Google Scholar 

  19. Graczyk, J., Swiatek, G.: Induced expansion for quadratic polynomials. Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, 399–482 (1996)

    Google Scholar 

  20. Graczyk, J., Swiatek, G.: Generic hyperbolicity in the logistic family. Ann. Math. 146, 1–52 (1997)

    MathSciNet  MATH  Google Scholar 

  21. Hayashi, S.: Connecting invariant manifolds and the solution of the C 1 stability and Ω-stability conjectures for flows. Ann. Math. 145, 81–137 (1997)

    MathSciNet  MATH  Google Scholar 

  22. Hubbard, J.H.: Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz. In: Topological Methods in Modern Mathematics, A Symposium in Honor of John Milnor’s 60th Birthday, 467–511. Publish or Perish 1993

    Google Scholar 

  23. Jakobson, M.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81, 39–88 (1981)

    MathSciNet  MATH  Google Scholar 

  24. Kozlovski, O.S.: Axiom A maps are dense in the space of unimodal maps in the C k topology. Ann. Math. 157, 1–43 (2003)

    MATH  Google Scholar 

  25. Kozlovski, O.S.: Getting rid of the negative Schwarzian derivative condition. Ann. Math. 152, 743–762 (2000)

    MathSciNet  MATH  Google Scholar 

  26. Lehto, O., Virtanen K.I.: Quasiconformal mappings in the plane. Springer 1973

  27. Levin, G., van Strien, S.: Local connectivity of Julia sets of real polynomials. Ann. Math. 147, 471–541 (1998)

    MathSciNet  MATH  Google Scholar 

  28. Levin, G., van Strien, S.: Bounds for maps of an interval with one critical point of inflection type II. IHES/M/99/82. Invent. Math. 141, 399–465 (2000)

    Google Scholar 

  29. Lyubich, M.: Some typical properties of the dynamics of rational maps. Russ. Math. Surv. 38, 154–155 (1983)

    MATH  Google Scholar 

  30. Lyubich, M.: On the Lebesgue measure of the Julia set of a quadratic polynomial. Preprint IMS at Stony Brook, # 1991/10

  31. Lyubich, M.: Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. Math. 140, 347–404 (1994). Note on the geometry of generalized parabolic towers. Manuscript 2000 (available at http://www.arXiv.org)

    MathSciNet  MATH  Google Scholar 

  32. Lyubich, M.: Dynamics of quadratic polynomials, I–II. Acta Math. 178, 185–297 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Lyubich, M.: Dynamics of quadratic polynomials, III. Parapuzzle and SBR measures. Astérisque 261, 173–200 (2000)

    Google Scholar 

  34. Lyubich, M.: Feigenbaum-Coullet-Tresser Universality and Milnor’s Hairiness Conjecture. Ann. Math. 149, 319–420 (1999)

    MathSciNet  MATH  Google Scholar 

  35. Lyubich, M.: Almost every real quadratic map is either regular or stochastic. Ann. Math. 156, 1–78 (2002)

    MathSciNet  MATH  Google Scholar 

  36. Lyubich, M., Yampolsky, M.: Dynamics of quadratic polynomials: Complex bounds for real maps. Ann. Inst. Fourier 47, 1219–1255 (1997)

    MathSciNet  MATH  Google Scholar 

  37. Mañé, R.: Hyperbolicity, sinks and measures for one-dimensional dynamics. Commun. Math. Phys. 100, 495–524 (1985)

    MathSciNet  Google Scholar 

  38. Martens, M.: Distortion results and invariant Cantor sets for unimodal maps. Ergodic Theory Dyn. Syst. 14, 331–349 (1994)

    MathSciNet  MATH  Google Scholar 

  39. Martens, M., de Melo, W., van Strien, S.: Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. 168, 273–318 (1992)

    MathSciNet  MATH  Google Scholar 

  40. Martens, M., Nowicki, T.: Invariant measures for Lebesgue typical quadratic maps. Astérisque 261, 239–252 (2000)

    MATH  Google Scholar 

  41. May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–466 (1976)

    CAS  PubMed  Google Scholar 

  42. McMullen, C.: Complex dynamics and renormalization. Ann. Math. Stud. 135 (1994)

  43. McMullen, C.: Renormalization and 3-manifolds which fiber over the circle. Ann. Math. Stud. 142 (1996)

  44. Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. Publ. Math., Inst. Hautes Étud. Sci. 53, 17–51 (1981)

    Google Scholar 

  45. Mañé, R., Sad, P., Sullivan, D.: On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 193–217 (1983)

    Google Scholar 

  46. de Melo W., van Strien S.: One-dimensional dynamics. Springer 1993

  47. Milnor, J., Thurston, W.: On iterated maps of the interval. Dynamical Systems. Proc. U. Md., 1986–87, ed. J. Alexander. Lect. Notes Math. 1342, 465–563 (1988)

  48. Newhouse, S., Palis, J., Takens, F.: Bifurcation and stability of families of diffeomorphisms. Publ. Math., Inst. Hautes Étud. Sci. 57, 5–71 (1983)

    Google Scholar 

  49. Nowicki, T.: A positive Liapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity. Ergodic Theory Dyn. Syst. 8, 425–435 (1988)

    MathSciNet  MATH  Google Scholar 

  50. Nowicki, T., van Strien, S.: Invariant measures exist under a summability condition for unimodal maps. Invent. Math. 105, 123–136 (1991)

    MathSciNet  MATH  Google Scholar 

  51. Palis, J.: A global view of dynamics and a conjecture of the denseness of finitude of attractors. Astérisque 261, 335–347 (2000)

    MATH  Google Scholar 

  52. Pugh, C.: The closing lemma. Am. J. Math. 89, 956–1009 (1967)

    MATH  Google Scholar 

  53. Rudin, W.: Real and complex analysis, second edition. McGraw-Hill 1974

  54. Sario, L., Nakai, M.: Classification theory of Riemann surfaces. Springer 1970

  55. Shishikura, M.: Topological, geometric and complex analytic properties of Julia sets. In: Proceedings of the International Congress of Mathematicians (Zürich, 1994), pp. 886–895. Basel: Birkhäuser 1995

  56. Singer, D.: Stable orbites and bifurcations of maps of the interval. SIAM J. Appl. Math. 35, 260–267 (1978)

    MATH  Google Scholar 

  57. Sullivan, D., Thurston, W.: Extending holomorphic motions. Acta Math. 157, 243–257 (1986)

    MathSciNet  MATH  Google Scholar 

  58. Slodkowski, Z.: Holomorphic motions and polynomial hulls. Proc. Am. Math. Soc. 111, 347–355 (1991)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Collège de France, 3, rue d’Ulm, 75005, Paris, France

    Artur Avila

  2. Mathematics Department and IMS, SUNY Stony Brook, 11794, Stony Brook, NY, USA

    Mikhail Lyubich

  3. Department of Mathematics, University of Toronto, M5S 3G3, Toronto, Ontario, Canada

    Mikhail Lyubich

  4. Instituto Nacional Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

    Welington de Melo

Authors
  1. Artur Avila
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikhail Lyubich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Welington de Melo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Artur Avila, Mikhail Lyubich or Welington de Melo.

Additional information

To Jacob Palis on his 60th birthday

Mathematics Subject Classification (2000)

37E05, 37F45, 30D05

Rights and permissions

Reprints and permissions

About this article

Cite this article

Avila, A., Lyubich, M. & de Melo, W. Regular or stochastic dynamics in real analytic families of unimodal maps. Invent. math. 154, 451–550 (2003). https://doi.org/10.1007/s00222-003-0307-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-003-0307-6

Keywords

Search

Navigation