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Invariant Measures Exist Without a Growth Condition

Bruin, H., Shen, W. and van Strien, S. (2003) Invariant Measures Exist Without a Growth Condition Communications in Mathematical Physics, 241. pp. 287-306.

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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>Df<sup>n</sup> (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>

Item Type: Article
Additional Information: This is a pre-press version of an article published in Communications in Mathematical Physics, 241, 287-306. Click here to access the published version. © 2003 Springer.
Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Mr Adam Field
Date Deposited: 27 May 2010 14:41
Last Modified: 23 Sep 2013 18:33
URI: http://epubs.surrey.ac.uk/id/eprint/1520

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