Convergence of moments for Axiom A and non-uniformly hyperbolic flows

2014

Electron. J. Probab.

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Section: Statement Of Resultsmentioning

confidence: 99%

“…[MT12b] showed that convergence of all moments holds also for nonuniformly expanding/hyperbolic diffeomorphisms modelled by Young towers with exponential tails [You98]. However, it follows from [MN08,MT12b] that the situation is quite different for systems modelled by Young towers with polynomial tails [You99].…”

Section: Statement Of Resultsmentioning

confidence: 99%

Moment bounds and concentration inequalities for slowly mixing dynamical systems

Gouëzel¹,

2014

Electron. J. Probab.

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“…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].…”

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confidence: 83%

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confidence: 99%

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

Bálint

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...

“…By the triangle inequality, it follows from (3.4) and (3.5) that The bound on moments in Proposition 3.8 together with the distributional limit law in Proposition 3.7 implies convergence of lower moments, (see for example[21]), and so lim n→∞ n X×S 1 ) = S Y (ω)/r. Now n−1 j=0 u(f j ω (x, ϕ)) = e iϕ n−1 j=0 e ijω v(f j (x)) and so | n−1j=0 u • f j ω | L 2 (X×S 1 ) = | n−1 j=0 e ijω v • f j | L 2 (X) .…”

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confidence: 96%

Broadband nature of power spectra for intermittent maps with summable and nonsummable decay of correlations

Gottwald

J. Phys. A: Math. Theor.

2016

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We present results on the broadband nature of the power spectrum S(ω), ω ∈ (0, 2π), for a large class of nonuniformly expanding maps with summable and nonsummable decay of correlations. In particular, we consider a class of intermittent maps f : [0, 1] → [0, 1] with f (x) ≈ x 1+γ for x ≈ 0, where γ ∈ (0, 1). Such maps have summable decay of correlations when γ ∈ (0, 1 2 ), and S(ω) extends to a continuous function on [0, 2π] by the classical Wiener-Khintchine Theorem. We show that S(ω) is typically bounded away from zero for Hölder observables.Moreover, in the nonsummable case γ ∈ [ 1 2 , 1), we show that S(ω) is defined almost everywhere with a continuous extensionS(ω) defined on (0, 2π), and S(ω) is typically nonvanishing.More generally, the power spectrum is well-defined and continuous provided the autocorrelations are summable.The power spectrum is often used by experimentalists to distinguish periodic and quasiperiodic dynamics (discrete power spectrum with peaks at the harmonics and subharmonics) and chaotic dynamics (broadband power spectra). See for example [10]. In the atmospheric and oceanic sciences and in climate science, power spectra have been widely used to detect variability in particular frequency bands (see, for example, [9,7]). Spectral analysis was successful in detecting dominant time scales in teleconnection patterns, revealing intraseasonal variability in time series of the global atmospheric angular momentum [6], interannual variability in the El Niño/Southern Oscillation system [4,14], and the Atlantic Multidecadal Variability [5]. On millenial temporal scales the power spectrum was instrumental in unraveling dominant cycles in paleoclimatic records [25,26].Despite this widespread applicability across these disparate temporal scales, there are surprisingly few rigorous results on the nature of power spectra of complex systems. The (quasi)periodic case with its peaks at discrete frequencies is well understood. The nature of power spectra for chaotic systems was first treated in [23] in the case of uniformly hyperbolic (Axiom A) systems. In our previous paper [16], we considered in more detail the broadband nature of power spectra for chaotic dynamical systems and showed that for certain classes of dynamical systems f and observables v, the power spectrum is bounded away from zero.The main results in [16] are for nonuniformly expanding/hyperbolic dynamical systems with exponential decay of correlations. These results are summarised below in Subsection 1.1. The current paper is concerned with systems possessing subexponential -even nonsummable -decay of correlations. A prototypical example is the class of Pomeau-Manneville intermittent maps [22], specifically the class considered in [15]. These are maps f : [0, 1] → [0, 1] given by