Melbourne, Ian and Terhesiu, Dalia (2012) Operator renewal theory and mixing rates for dynamical systems with infinite measure. Inventiones mathematicae, Vol. 189 (No. 1). pp. 61-110. doi:10.1007/s00222-011-0361-4 ISSN 0020-9910.

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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 ∑nj=1Lj ) 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:	Journal Article
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Inventiones mathematicae
Publisher:	Springer
ISSN:	0020-9910
Official Date:	2012
Dates:	Date Event 2012 Published
Volume:	Vol. 189
Number:	No. 1
Page Range:	pp. 61-110
DOI:	10.1007/s00222-011-0361-4
Status:	Peer Reviewed
Publication Status:	Published

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