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

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An Erratum to this article was published on 12 August 2015

Abstract

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 \(\sum_{j=1}^{n}L^{j}\)) 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK

    Ian Melbourne & Dalia Terhesiu

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  1. Ian Melbourne
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  2. Dalia Terhesiu
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Correspondence to Ian Melbourne.

Additional information

An erratum to this article is available at http://dx.doi.org/10.1007/s00222-015-0616-6.

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Melbourne, I., Terhesiu, D. Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. math. 189, 61–110 (2012). https://doi.org/10.1007/s00222-011-0361-4

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  • DOI: https://doi.org/10.1007/s00222-011-0361-4

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