Communications in Mathematical Physics

, Volume 241, Issue 2, pp 287–306

Invariant Measures Exist Without a Growth Condition


DOI: 10.1007/s00220-003-0928-z

Cite this article as:
Bruin, H., Shen, W. & Strien, S. Commun. Math. Phys. (2003) 241: 287. doi:10.1007/s00220-003-0928-z


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 |Dfn(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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Henk Bruin
    • 1
  • Weixiao Shen
    • 2
  • Sebastian van Strien
    • 2
  1. 1.Department of MathematicsUniversity of GroningenAV GroningenThe Netherlands
  2. 2.Department of MathematicsUniversity of WarwickCoventry CV4 7ALUK