Existence of unique SRB-measures is typical for real unicritical polynomial families

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Abstract

We show that for a one-parameter family of unicritical polynomials {fc} with even critical order ℓ⩾2, for almost all parameters c, fc admits a unique SRB-measure, being either absolutely continuous, or supported on the postcritical set. As a byproduct we prove that if fc has a Cantor attractor, then it is uniquely ergodic on its postcritical set.

Résumé

Nous montrons que si {fc} est une famille à un paramètre de polynômes unicritiques dont l'ordre ℓ⩾2 est pair, alors pour presque toute valeur du paramètre c, fc admet une unique mesure SRB et soit cette mesure est absolument continue, soit son support est l'ensemble postcritique. Nous montrons aussi que, si fc a un attracteur de Cantor, alors fc est uniquement ergodique.

References (39)

  • H. Bruin et al.

    Decay of correlations in one-dimensional dynamics

    Ann. Sci. École Norm. Sup.

    (2003)
  • L. Ahlfors

    Lectures on Quasiconformal Mappings

    (1966)
  • A. Avila et al.

    Regular or stochastic dynamics in real analytic families of unimodal maps

    Invent. Math.

    (2003)
  • A. Avila et al.

    Statistical properties of unimodal maps: The quadratic family

    Ann. of Math.

    (2005)
  • G. Birkhoff

    Extensions of Jentzsch's theorem

    Trans. Amer. Math. Soc.

    (1957)
  • A. Blokh et al.

    Measurable dynamics of S-unimodal maps of the interval

    Ann. Sci. École Norm. Sup.

    (1991)
  • H. Bruin

    Minimal Cantor systems and unimodal maps

    J. Difference Eq. Appl.

    (2003)
  • H. Bruin

    Topological conditions for the existence of Cantor attractors

    Trans. Amer. Math. Soc.

    (1998)
  • H. Bruin et al.

    Wild Cantor attractors exist

    Ann. of Math.

    (1996)
  • H. Bruin et al.

    Invariant measures exist without a growth condition

    Commun. Math. Phys.

    (2003)
  • Douady A., Hubbard J., Dynamical study of complex polynomials, Part I and Part II, Mathematical Publications of Orsay...
  • A. Douady et al.

    On the dynamics of polynomial-like mappings

    Ann. Sci. École Norm. Sup. (4)

    (1985)
  • F. Durand

    Linearly recurrent subshifts have a finite number of non-periodic subshift factors

    Ergodic Theory Dynam. Syst.

    (2000)
  • J. Graczyk et al.

    Induced expansion for quadratic polynomials

    Ann. Sci École Norm. Súp.

    (1996)
  • B. Hasselblatt et al.

    A First Course in Dynamics, with a Panorama of Recent Developments

    (2003)
  • F. Hofbauer et al.

    Some remarks on recent results about S-unimodal maps

    Ann. Inst. Henri Poincaré

    (1990)
  • Kahn J., Holomorphic removability of Julia sets, IMS preprint...
  • O. Kozlovski

    Getting rid of the negative Schwarzian derivative condition

    Ann. of Math.

    (2000)
  • Kozlovski O., Shen W., van Strien S., Rigidity for real polynomials, Ann. of Math., in...
    • 1

    Supported by EPSRC Grant GR/S91147/01.

    2

    Supported by EPSRC Grant GR/R73171/01 and Bai Ren Ji Hua program of the CAS.

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