Polynomial Decay of Correlations for Flows, Including Lorentz Gas Examples

Bálint, Péter; Butterley, Oliver; Melbourne, Ian

doi:10.1007/s00220-019-03423-6

Cited by 12 publications

(40 citation statements)

References 36 publications

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“…) in[12] follow from Propositions 4.1 and 4.2. Moreover, ϕ is constant along stable leaves by Proposition 9.1 and projects to a well-defined roof function ϕ : Y → R + satisfying the estimate in Proposition 9.4 which is condition (3.3) in…”

mentioning

confidence: 70%

“…We are now in a position to apply [ [12]. Hence the suspension flow on Y ϕ is a skew product Gibbs-Markov flow in the terminology of [12].…”

Section: Statistical Properties Of Singular Hyperbolic Attractorsmentioning

confidence: 99%

“…We are now in a position to apply [ [12]. Hence the suspension flow on Y ϕ is a skew product Gibbs-Markov flow in the terminology of [12]. Hence superpolynomial mixing follows from [12, Theorem 3.1] subject to a nondegeneracy condition (absence of approximate eigenfunctions).…”

Section: Statistical Properties Of Singular Hyperbolic Attractorsmentioning

confidence: 99%

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Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation

Araújo

Melbourne

2019

Advances in Mathematics

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Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the classical Lorenz attractor. However, many of the proofs rely heavily on the smoothness of the stable foliation for the flow.In this paper, we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the stable foliation. These properties include existence of SRB measures, central limit theorems and associated invariance principles, as well as results on mixing and rates of mixing. The properties hold equally for singular hyperbolic flows in higher dimensions provided the center-unstable subspaces are two-dimensional.

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mentioning

confidence: 70%

“…We are now in a position to apply [ [12]. Hence the suspension flow on Y ϕ is a skew product Gibbs-Markov flow in the terminology of [12].…”

Section: Statistical Properties Of Singular Hyperbolic Attractorsmentioning

confidence: 99%

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Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation

Araújo

Melbourne

2019

Advances in Mathematics

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“…By [18,19] it is therefore possible to prove homogenisation results where fast-slow systems converge to a limiting stochastic differential equation.Remark 1.1 One version of an earlier preprint of this paper was called "Decay of correlations for flows with unbounded roof function, including the infinite horizon planar periodic Lorentz gas". The correlation decay aspects of that preprint were incorrect and are subsumed by [10]. The remainder of that preprint is the basis of the current paper.We note also that a "rearrangement" argument plays a much larger role in the older versions, but is still useful for optimal moment control when the L p class of the…”

mentioning

confidence: 98%

“…Remark 1.1 One version of an earlier preprint of this paper was called "Decay of correlations for flows with unbounded roof function, including the infinite horizon planar periodic Lorentz gas". The correlation decay aspects of that preprint were incorrect and are subsumed by [10]. The remainder of that preprint is the basis of the current paper.…”

mentioning

confidence: 98%

Statistical Properties for Flows with Unbounded Roof Function, Including the Lorenz Attractor

Bálint

Melbourne

2018

J Stat Phys

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For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors.We also obtain the functional central limit theorem and moment estimates, as well as iterated versions of these results. A consequence is deterministic homogenisation (convergence to a stochastic differential equation) for fast-slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two. such as expansivity [9], the central limit theorem (CLT) [17], moment estimates [28], and exponential decay of correlations [6].Recently, [31] (see also previous work of [14]) gave an analytic proof of existence of geometric Lorenz attractors in the extended Lorenz modelA consequence of the current paper, in conjunction with [8], is that CLTs and moment estimates hold for these attractors. Exponential decay of correlations remains an open question: the proof in [6] relies on the existence of a smooth stable foliation for the flow, which holds for the classical Lorenz attractor [7] but not for the examples of [14,31]. (As far as the results on the CLT and moment estimates go, our results for the extended Lorenz model are completely analytic, whereas results for the classical Lorenz attractor rely on the computer-assisted proof in [35].)More generally, our results apply to all singular hyperbolic attractors for threedimensional flows. By [30], this incorporates all nontrivial robustly transitive attractors for three-dimensional flows (including nontrivial Axiom A attractors as well as classical and geometric Lorenz attractors). Our results apply also to codimension two singular hyperbolic attractors in arbitrary dimension.A standard step in the analysis of Lorenz attractors and singular hyperbolic attractors is to consider a suitable Poincaré map with good nonuniform hyperbolicity properties. The return time function (roof function) is generally unbounded due to the presence of steady-states for the flow. Due to this unboundedness, the original proof of the CLT for the classical Lorenz equations [17] is quite complicated, relying on an inducing scheme that involves phase-space exclusion arguments of the type in [12] to remove points that return too quickly to a vicinity of the steady-states.As we pointed out in an earlier unpublished preprint version of this paper, it is possible to give a much simpler proof of the CLT than the one in [17]. In this paper, we provide the details. As in [17], we prove also the functional version, namely the weak invariance principle (WIP). We also provide moment estimates as advertised in [28, Section 5.3] and our results apply to the general class of singular hyperbolic attractors in [8] (it is not clear that the argument in [17] applies in this generality). In addition, we prove iterated versions of the WIP and moment estimates as advertised in [18, Section 10.3]. By [18,19] it is therefore possible to prove homogenisation results wh...

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On Sinaĭ Billiards on Flat Surfaces with Horns

Bruin

2021

J Stat Phys

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We show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers (Ann. Math. 147:585–650, 1998) with exponential tails. This implies exponential decay of correlations for the billiard map. Because the height function of the suspension flow itself is polynomial when the horns are Torricelli-like trumpets, one can derive Limit Laws for the billiard flow, including Stable Limits if the parameter of the Torricelli trumpet is chosen in (1, 2).

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Polynomial Decay of Correlations for Flows, Including Lorentz Gas Examples

Abstract: We prove sharp results on polynomial decay of correlations for nonuniformly hyperbolic flows. Applications include intermittent solenoidal flows and various Lorentz gas models including the infinite horizon Lorentz gas.

Cited by 12 publications

References 36 publications

Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation

Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation

Statistical Properties for Flows with Unbounded Roof Function, Including the Lorenz Attractor

On Sinaĭ Billiards on Flat Surfaces with Horns

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