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Invariant Measures Exist Without a Growth Condition

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Abstract

Given a non-flat S-unimodal interval map f, we show that there exists C which only depends on the order of the critical point c such that if |Df n(f(c))|≥C for all n sufficiently large, then f admits an absolutely continuous invariant probability measure (acip). As part of the proof we show that if the quotients of successive intervals of the principal nest of f are sufficiently small, then f admits an acip. As a special case, any S-unimodal map with critical order ℓ<2+ɛ having no central returns possesses an acip. These results imply that the summability assumptions in the theorems of Nowicki & van Strien [21] and Martens & Nowicki [17] can be weakened considerably.

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Authors and Affiliations

  1. Department of Mathematics, University of Groningen, 800, 9700, AV Groningen, The Netherlands

    Henk Bruin

  2. Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK

    Weixiao Shen & Sebastian van Strien

Authors
  1. Henk Bruin
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  2. Weixiao Shen
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  3. Sebastian van Strien
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Corresponding author

Correspondence to Henk Bruin.

Additional information

Communicated by P. Sarnak

HB was supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences (KNAW)

WS was supported by EPSRC grant GR/R73171/01

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Bruin, H., Shen, W. & Strien, S. Invariant Measures Exist Without a Growth Condition. Commun. Math. Phys. 241, 287–306 (2003). https://doi.org/10.1007/s00220-003-0928-z

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  • DOI: https://doi.org/10.1007/s00220-003-0928-z

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