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Proceedings of the London Mathematical Society Advance Access published online on June 13, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn028
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© 2008 London Mathematical Society

Decay of correlations for slowly mixing flows

Ian Melbourne

Department of Mathematics
University of Surrey
Guildford
GU2 7XH
United Kingdom

Received 6 November 2006. Revision received 9 April 2008.

We show that polynomial decay of correlations is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of a class of nonuniformly hyperbolic diffeomorphisms for which Young proved polynomial decay of correlations. Roughly speaking, in situations where the decay rate O(1/nβ) has previously been proved for diffeomorphisms, we establish the decay rate O(1/tβ) for flows. Applications include certain classes of semidispersing billiards, as well as dispersing billiards with vanishing curvature. In addition, we obtain results for suspension flows with unbounded roof functions. In particular, the classical planar Lorentz flow with a doubly periodic array of circular scatterers has decay rate 1/t as anticipated by physicists.

2000 Mathematics Subject Classification 37A25, 37D25, 37D50.

This research was supported in part by EPSRC grant EP/D055520/1 and by a Leverhulme Research Fellowship.


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