Superpolynomial and polynomial mixing for semiflows and flows

Abstract

We give a review of results on superpolynomial decay of correlations, and polynomial decay of correlations for nonuniformly expanding semiflows and nonuniformly hyperbolic flows. A self-contained proof is given for semiflows. Results for flows are stated without proof (the proofs are contained in separate joint work with Bálint and Butterley). Applications include intermittent solenoidal flows, suspended Hénon attractors, Lorenz attractors and singular hyperbolic attractors, and various Lorentz gas models including the infinite horizon Lorentz gas. Contents

Polynomial Decay of Correlations for Flows, Including Lorentz Gas Examples

“…See also Remark 8.5. Here we give a different proof that has a number of advantages as discussed in the introduction to [26]. Flows are modelled as suspensions over a uniformly hyperbolic map with an unbounded roof function (rather than as suspensions over a nonuniformly hyperbolic map with a bounded roof function).…”

“…Examples covered by our results on rapid mixing include finite Lorentz gases (including those with cusps, corner points, and external forcing), Lorenz attractors, and Hénon-like attractors. We refer to [26] for references and further details.…”

“…These aspects are completed and extended in [7].) (b) A drawback of the method in this paper, already present in [18] and inherited by [24,25,26], is that at least one of the observables v or w is required to be C m in the flow direction. Here m can be estimated, with difficulty, but is likely to be quite large.…”

“…In Part II, we show how these results apply to important classes of nonuniformly hyperbolic flows including those mentioned in this introduction. The methods of proof in this paper, especially those in Part I, are fairly straightforward adaptations of those in [26]. The main new contribution of the paper (Section 6 together with Part II) is to develop a general framework whereby large classes of nonuniformly hyperbolic flows, including fundamental examples such as the infinite horizon Lorentz gas, are covered by these methods.…”

“…2 The paper has been structured to be as self-contained as possible. It does not seem possible to reduce the results on flows in Part I of this paper to the results on semiflows in [26]. Instead, it is necessary to start from scratch and to emulate, rather than apply directly, the methods in [26].…”