Variance Continuity for Lorenz Flows

The classical Lorenz flow, and any flow which is close to it in the C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document}-topology, satisfies a Central Limit Theorem (CLT). We prove that the variance in the CLT varies continuously.


Introduction
In 1963, Lorenz [20] introduced the following system of equations: as a simplified model for atmospheric convection. Numerical simulations performed by Lorenz showed that the above system exhibits sensitive dependence on initial conditions and has a non-periodic "strange" attractor. Since then, (1) became a basic example of a chaotic deterministic system that is notoriously difficult to analyse. A rigorous mathematical framework of similar flows was initiated with the introduction of the so-called geometric Lorenz attractors in [1,15]. The papers [24,25] provided a computer-assisted proof that the classical Lorenz attractor in (1) is indeed a geometric Lorenz attractor. In particular, it is a singularly hyperbolic attractor [23], namely a nontrivial robustly transitive attracting invariant set containing a singularity (equilibrium point). Moreover, there is a distinguished Sinai-Ruelle-Bowen (SRB) ergodic probability measure; see WB and MR would like to thank The Leverhulme Trust for supporting their research through the research Grant RPG-2015-346. The research of IM was supported in part by a European Advanced Grant StochExtHomog (ERC AdG 320977). We are very grateful to the referee for important comments that have led to a much more readable explanation of the method in this paper.
for example [8]. Statistical limit laws, in particular the central limit theorem (CLT) for Hölder observables, were first obtained in [16] for the classical Lorenz attractor and were shown for general singular hyperbolic attractors in [5]. For further background and a complete list of references up to 2010, we refer the reader to the monograph of Araújo and Pacífico [7].
Let X ε : R 3 × R → R 3 , ε ≥ 0, be a continuous family of C 2 flows on R 3 admitting a geometric Lorenz attractor with singularities x ε and corresponding SRB measures μ ε . Precise definitions are given in Sect. 2.1; in particular, the framework includes the classical Lorenz attractor. By [2,9], the flows X ε are statistically stable: for any continuous ψ: The CLT in [5,16] states that for fixed ε ≥ 0 and ψ:R 3 → R Hölder, there By [16,Section 4.3], σ 2 is typically nonzero. We prove continuity of the variance, namely that ε → σ 2 Xε (ψ) is continuous. At the same time, we obtain estimates on the dependence of the variance on ψ. We now state the main result of the paper. Define ψ = |ψ| dμ 0 + |ψ(x 0 )|.
In part (b), we obtain closeness of the variances provided the observables are close in L 1 with respect to both the SRB measure and the Dirac point mass at 0 (provided the individual Hölder norms are controlled). It is an easy consequence of the methods in this paper that the logarithmic factor can be removed if the norm is replaced by the Hölder norm.
The paper is organised as follows. In Sect. 2, we recall the basic set-up and notation associated with (families of) geometric Lorenz attractors. In Sect. 3, we show how to normalise the families of flows to obtain simplified coordinates for the proofs. Section 4 contains properties of one-dimensional Lorenz maps. Section 5 studies the family of Poincaré maps. It starts by showing that the family of maps admit a uniform rate of correlations decay for piecewise Hölder functions, using suitable anisotropic norms. We then use the Green-Kubo formula to show continuity of the variance for the family of Poincaré maps. In Sect. 6, we prove a version of Theorem 1.1 for normalised families and use this to prove Theorem 1.1.

Geometric Lorenz Attractors
In this section, we recall the basic set-up and notation associated with (families of) geometric Lorenz attractors. In Sects. 2.1 and 2.2, we recall the class of (families of) geometric Lorenz attractors considered in the paper.
We begin with some notational preliminaries. Let U ⊂ R m be open. Fix α ∈ (0, 1) and recall that f : Similarly, we speak of continuous families of C 2 flows, C 1+α diffeomorphisms, and so on. In the case of Lorenz flows, we are particularly interested in families of C 2 flows on R 3 restricted to an open-bounded region U of phase space; for convenience, we suppress mentioning the subset U .

Definition of Geometric Lorenz Attractors
There are various notions of geometric Lorenz attractor in the literature depending on the properties being analysed. Roughly speaking, we take a geometric Lorenz attractor to be a singular hyperbolic attractor for a vector field on R 3 with a single singularity x 0 and a connected global cross section with a C 1+α stable foliation. As promised, we now give a precise description.
Let Σ be a two-dimensional rectangular cross section transverse to the flow chosen in a neighbourhood of the singularity x 0 , and let Γ be the intersection of Σ with the local stable manifold of x 0 . We suppose that there exists a well-defined Poincaré map F :Σ\Γ → Σ. Moreover, we assume that the underlying flow is singular hyperbolic [23]. It follows [4,Theorem 4.2] that a neighbourhood of the attractor is foliated by one-dimensional C 2 stable leaves. We assume q-bunching for some q > 1 in [4, condition (4.2)]. By [4,Theorem 4.12 and Remark 4.13(b)], it follows that the stable foliation for the flow is C q .
The foliation by stable leaves for the flow naturally induces (see for example [5, Section 3.1]) a C q (q > 1) foliation inside Σ of a neighbourhood of the attractor intersected with Σ by one-dimensional C 2 stable leaves for the Poincaré map F . We denote this stable foliation for F by F.
The stable leaves for the flow are exponentially contracting [4, Theorem 4.2(a)(3)], and this property is inherited by the stable leaves for F . This means that there exists ρ ∈ (0, 1), K > 0 such that for ξ 1 , ξ 2 in the same stable leaf in F and n ≥ 1.
Let I ⊂ Σ be a smoothly (C ∞ ) embedded one-dimensional subspace transverse to the stable foliation, and let T :I → I be the one-dimensional map obtained from F by quotienting along stable leaves. Let ξ 0 be the intersection of I with Γ.
Proposition 2.1. T is a Lorenz-like expanding map. That is, T is monotone (without loss we take T to be increasing) and piecewise C 1+α on I\{ξ 0 } for some α ∈ (0, 1) with a singularity at ξ 0 and one-sided limits T (ξ + 0 ) < 0 and T (ξ − 0 ) > 0. Also, T is uniformly expanding: there are constants c > 0 and Proof. The map F is piecewise C 1+α , and the foliation by stable leaves is C 1+α , so T is piecewise C 1+α on I\{ξ 0 }. Uniform expansion follows from [5,Theorem 4.3]. The remaining properties are immediate.
The final part of the definition of geometric Lorenz attractor is that the one-dimensional map T is transitive on the interval [T (ξ + 0 ), T (ξ − 0 )]. It is then standard [2,5,8,17] that T has a unique absolutely continuous invariant probability measure (acip)μ leading to a unique SRB measure μ for the geometric Lorenz attractor containing x 0 . The basin of μ has full Lebesgue measure in a neighbourhood of the attractor.
Remark 2.2. The classical Lorenz attractor for the system of equations (1) (and for nearby equations) is an example of a geometric Lorenz attractor as defined above. Except for q-bunching, the assumptions above are verified in [25]. The q-bunching condition is verified in [6, Lemma 2.2]. (By [4, Section 5], the optimal value of q lies between 1.278 and 1.705; hence, we have C 1+α regularity for the stable foliation as in [9] but not C 2 regularity as in [2].)

Families of Geometric Lorenz Attractors
Let X ε , ε ≥ 0, be a continuous family of C 2 flows on R 3 admitting a geometric Lorenz attractor as in Sect. 2.1 with singularity x ε . The constants K and ρ in (3) derive from the singular hyperbolic structure which varies continuously under C 1 perturbations. Hence, K and ρ can be chosen independent of ε.
Making an initial C 2 change of coordinates (varying continuously in ε), we can suppose without loss that x ε ≡ 0 and that Σ, Γ and I are given by 1 2 ] for all ε. Throughout the paper, when we speak of a continuous family of C 2 flows admitting geometric Lorenz attractors, we assume that this initial change of coordinates has been performed.
Define the Poincaré return time to Σ, Applying the Hartman-Grobman theorem for fixed ε, we can linearise X ε in a neighbourhood of 0 so that the linearised flow is given by (x, y, z) → (e λ1,εt x, e λ2,εt y, e λ3,εt z). The time of flight in this neighbourhood is readily calculated in these coordinates to be − 1 λ1,ε log |x| for x ∈ I and the same formula holds in the original coordinates. The remaining flight time τ 2,ε is a first hit time for the C 2 flow away from the singularity at 0 and hence is C 2 . Since X ε is a continuous family of C 2 flows, it follows that τ 2,ε is a continuous family of C 2 functions. Theorem 2.4. Let X ε be a continuous family of C 2 flows on R 3 admitting a geometric Lorenz attractor. Then, there exists α > 0 such that the onedimensional maps T ε :I → I form a continuous family of piecewise C 1+α maps.
Proof. Recall that T ε is obtained from the continuous family of piecewise C 1+α maps F ε by quotienting along the stable foliation. Our assumption of qbunching (q > 1) yields continuous families of C q stable foliations [12]. Hence, T ε is a continuous family of piecewise C 1+α maps.

Normalised Geometric Lorenz Attractors
Let X ε , ε ≥ 0, be a continuous family of C 2 flows on R 3 admitting a geometric Lorenz attractor. In this section, we show how to normalise the families of flows to obtain simplified coordinates for carrying out the proofs.
Assume that the preliminary C 2 change of coordinates in Sect. 2.2 has been performed. Let F ε :Σ → Σ and T ε :I → I be the associated families of Poincaré maps and one-dimensional piecewise expanding maps. Also, define Proof. For ε fixed, this follows by definition of the smoothness of the stable foliation for F ε . (An explicit formula is given in [4,Lemma 4.9] where Y and χ should be replaced by I and ω ε , and the embedding is the identity.) Again, we obtain continuous families of C 1+α diffeomorphisms by [12].
The change of coordinates ω ε for the Poincaré map F ε extends to a change of coordinates φ ε for the flow X ε . The extension is standard and essentially unique, though heavy on notation.
First, define the transformed Poincaré map and return timẽ Hence, φ ε is the desired change of coordinates. Proof. By assumption, X ε is a continuous family of C 2 flows. Hence, the result follows from Proposition 3.1.
diffeomorphisms and has the form where T ε :I → I is the family of Lorenz-like expanding maps in Theorem 2.4(a). y)). By Proposition 3.1(a), the maps T ε :I → I are unchanged by this change of coordinates. Also, ω ε is a continuous family of C 1+α diffeomorphisms and F ε is a continuous family of piecewise C 1+α diffeomorphisms, soF ε a continuous family of piecewise C 1+α diffeomorphisms.
(iv) By Proposition 3.1(a), the acipμ ε for T ε is unchanged by the change of coordinates and hence remains absolutely continuous. Using this and the construction of the SRB measure (see for example [2,8] for the standard construction of μ ε fromμ ε ), we obtain thatμ ε = φ −1 ε * μ ε is the SRB measure forX ε . Moreover, strong statistical stability [2,9,14] of the acips μ ε on I is preserved and hence the SRB measuresμ ε remain statistically stable.
In the following sections, we prove Theorem 1.1 for normalised families of geometric Lorenz attractors. At the end of Sect. 6, we show how results for normalised families imply Theorem 1.1.

The Family of One-Dimensional Maps
In this section, we recall some functional-analytic properties associated with the family of one-dimensional Lorenz maps T ε . For p ≥ 1, we say f :I → R is of (universally) bounded p-variation if where the essential supremum is taken with respect to Lebesgue measure on I × I and · 1 is the L 1 -norm with respect to Lebesgue measure on I. Fix ρ 0 > 0 and let BV 1,1/p ⊂ L 1 be the Banach space equipped with the norm (The space BV 1,1/p does not depend on ρ 0 .) The fact that BV 1,1/p is a Banach space is proved in [17]. Moreover, it is proved in [17] that BV 1,1/p is embedded in L ∞ and compactly embedded in L 1 . In addition, [17] shows that We recall results from the literature that we use later in Sects. 5 and 6of the paper. Recall from [17] that T ε admits a unique acipμ ε for each ε ≥ 0. Let h ε denote the density forμ ε . Let P ε : L 1 (I) → L 1 (I) denote the transfer operator (Perron-Frobenius) associated with T ε (so P ε f g d Leb = f g • T ε d Leb for f ∈ L 1 (I), g ∈ L ∞ (I)).

Variance Continuity for the Poincaré Maps
In this section, we prove the analogue of Theorem 1.1 at the level of the Poincaré maps F ε . Throughout, we work with normalised families as in Sect. 3.
It is well known [8] that F ε admits a unique SRB measure μ Fε for each ε ≥ 0. Moreover, for any continuous ψ : i.e. the family F ε is statistically stable [2,9,14]. We require the following strengthening of this property. In general, we say that Ψ : Σ → R is piecewise continuous if it is uniformly continuous on the connected components of Σ\Γ. Similarly, Ψ is piecewise Hölder if it is uniformly Hölder on the connected components of Σ\Γ. Proposition 5.1. Suppose that Ψ:Σ → R is piecewise continuous and fix n ≥ 0.
Proof. If Ψ · (Ψ • F n 0 ) were continuous, this would be immediate from the statement of Proposition 3.3 in [2]. The proof in [2] already accounts for trajectories that visit Γ in finitely many steps, and it is easily checked that the same arguments apply here.

Theorem 5.2. Assume that there exists K > 0 such that
for all j ≥ 1, ε ≥ 0. Then, there exist C > 0 and θ ∈ (0, 1) such that Proof. The proof follows from [3, Theorem 3]. The fact that C and θ do not depend on ε follows from the uniformity of the constants in Proposition 4.1(b) and the uniformity of the contraction on vertical fibres.
Recall that F ε can be written in coordinates as F ε (x, y) = (T ε x, g ε (x, y)).
Proof. We suppress the dependence on ε.
(The hypotheses of this result will be verified in Sect. 6. In particular, condition (7) is addressed in Lemma 6.2.) We first prove a lemma that will be used in the proof of Theorem 5.6.
Lemma 5.7. Let Ψ:Σ → R be piecewise continuous and fix n ≥ 0. Then, The closure of E δ,n lies in the interior of E 2δ,n , so there exists a continuous function χ:Σ → [0, 1] supported in E 2δ,n and equal to 1 on E δ,n . By statistical stability, for ε sufficiently small, and all sufficiently small ε. Moreover, Ψ • F n−1 0 is uniformly continuous on E c δ/2,n−1 . It follows from this and the uniform convergence of F ε on E c δ,n that Hence, The result follows since δ > 0 is arbitrary.
We end this section with the following result which gives explicit estimates in terms of Ψ as required for the proof of Theorem 1.1(b).

Variance Continuity for the Flows
By [5,16], the CLT holds for Hölder observables for the Lorenz flows X ε . In this section, we show how to obtain continuity of the variances, proving Theorem 1.1. During most of this section, we continue to work with normalised families as in Sect. 3, culminating in Corollary 6.5 which is an analogue of Theorem 1.1 for normalised families. We conclude by using Corollary 6.5 to prove Theorem 1.1. Throughout, we fix β ∈ (0, 1).