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Operator renewal theory and mixing rates for dynamical systems with infinite measure

Melbourne, I and Terhesiu, D (2011) Operator renewal theory and mixing rates for dynamical systems with infinite measure Inventiones Mathematicae, 189 (1). 61 - 110. ISSN 0020-9910

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We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Date : 12 October 2011
Identification Number : 10.1007/s00222-011-0361-4
Related URLs :
Additional Information : The original publication is available at
Depositing User : Symplectic Elements
Date Deposited : 15 Dec 2011 13:51
Last Modified : 09 Jun 2014 13:22

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