Invariant Measures Exist Without a Growth Condition

Abstract

<p>Given a non-flat S-unimodal interval map <i>f</i>, we show that there exists <i>C</i> which only depends on the order of the critical point <i>c</i> such that if |<i>Dfⁿ (f(c)</i>)|greater than or equal to <i>C</i> for all <i>n</i> sufficiently large, then <i>f</i> 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 <i>f</i> are sufficiently small, then <i>f</i> admits an acip. As a special case, any S-unimodal map with critical order <i>l</i> < 2+ <i>epsilon</i> having no central returns possesses an acip. These results imply that the summability assumptions in the theorems of Nowicki &amp; van Strien [21] and Martens &amp; Nowicki [17] can be weakened considerably.</p>

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