, Volume 241, Issue 2-3, pp 287-306

Invariant Measures Exist Without a Growth Condition

Purchase on Springer.com

$39.95 / €34.95 / £29.95*

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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.

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