DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 13, Number 5, December 2005 pp. 1187–1201
ANOTHER PROOF OF THE AVERAGING PRINCIPLE FOR FULLY COUPLED DYNAMICAL SYSTEMS WITH HYPERBOLIC FAST MOTIONS
Yuri Kifer Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel
Abstract. In the study of systems which combine slow and fast motions which depend on each other (fully coupled setup) whenever the averaging principle can be justified this usually can be done only in the sense of L1-convergence on the space of initial conditions. When fast motions are hyperbolic (Axiom A) flows or diffeomorphisms (as well as expanding endomorphisms) for each freezed slow variable this form of the averaging principle was derived in [19] and [20] relying on some large deviations arguments which can be applied only in the Axiom A or uniformly expanding case. Here we give another proof which seems to work in a more general framework, in particular, when fast motions are some partially hyperbolic or some nonuniformly hyperbolic dynamical systems or nonuniformly expanding endomorphisms.
1. Introduction. Many real systems can be viewed as perturbations of an idealized system with integrals of motion which start changing slowly after a perturbation producing a combination of slow and fast motions which can only be described by complicated double scale equations whose analysis is not easy. Already in the 19th century in applications to celestial mechanics it was well understood (though without rigorous justification) that a good approximation of the slow motion can often be obtained by averaging its parameters in fast variables. Later, averaging methods were applied in other mechanical problems, in signal processing both in deterministic and stochastic setups and, rather recently, to model climate–weather interactions (see [13] and [18]). The classical setup of averaging justified rigorously in [6] presumes that the fast motion does not depend on the slow one and most of the work on averaging treats this case only which is, essentially, completely understood. On the other hand, in real systems both slow and fast motions depend on each other which leads to the more difficult fully coupled case which we study in this paper. This set up emerges, in particular, in typical perturbations of Hamiltonian systems where motions on constant energy manifolds are fast and across them are slow. ε ε ε Namely, we consider a system of differential equations for X = Xx,y and Y = ε Yx,y, dXε(t) dY ε(t) = εB(Xε(t),Y ε(t)), = b(Xε(t),Y ε(t)) (1.1) dt dt with initial conditions Xε(0) = x, Y ε(0) = y on the product Rd × M where M is a compact n−dimensional Riemannian manifold and B(x, y), b(x, y) are bounded and smooth in x, y vector fields on Rd and M, respectively. The solutions of
2000 Mathematics Subject Classification. Primary: 37D20 Secondary: 34C29, 60F10. Key words and phrases. averaging principle, hyperbolic attractors. The author was partially supported by US-Israel BSF.
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t d t (1.1) determine the flow of diffeomorphisms Φε on R × M acting by Φε(x, y) = ε ε t t t (Xx,y(t),Yx,y(t)). Taking ε = 0 we arrive at the flow Φ = Φ0 acting by Φ0(x, y) = t t t (x, Fxy) where Fx is another family of flows given by Fxy = Yx,y(t) with Y = Yx,y = 0 Yx,y being the solution of dY (t) = b(x, Y (t)),Y (0) = y. (1.2) dt It is natural to view the flow Φt as describing an idealized physical system where t parameters x = (x1, ..., xd) are assumed to be constant while the perturbed flow Φε is regarded as describing a real system where evolution of these parameters is also taken into consideration. t Let µx be an ergodic invariant measure of the flow Fx. Then the limit Z T ¯ ¯ −1 t B(x) = By(x) = lim T B(x, Fxy)dt (1.3) T →∞ 0 exists for µx−almost all y and it is equal to Z ¯ ¯ B(x) = Bµx (x) = B(x, y)dµx(y). (1.4)
t t If b(x, y) does not, in fact, depend on x then Fx = F and if µx = µ is also independent of x we arrive at the classical setup. In this case Lipschitz continuity of B implies already that B¯(x) is also Lipschitz continuous in x, and so there exists a unique solution X¯ = X¯x of the averaged equation dX¯ ε(t) = εB¯(X¯ ε(t)), X¯ ε(0) = x. (1.5) dt Then the classical averaging principle says (see [27]) that ε ¯ ε lim sup |Xx,y(t) − Xx(t)| = 0 (1.6) ε→0 0≤t≤T/ε for all x and for those y for which (1.3) holds true. In the general case (1.1) the averaged vector field B¯(x) in (1.3)–(1.4) may even not be continuous in x, let alone Lipschitz, and so (1.5) may have many solutions or not at all. Moreover, there d may exist no natural well dependent on x ∈ R family of invariant measures µx t since dynamical systems Fx may have rather different properties for different x’s. Even when all measures µx are the same the averaging principle often does not hold true in the form (1.6), for instance, in the presence of resonances (see [22]). Thus even basic results on approximation of the slow motion by the averaged one in the fully coupled case should be formulated usually in a different way and they require stronger and more specific assumptions. If convergence in (1.3) is uniform in x and y then (see, for instance, [21]) any limit ¯ ¯ ε ε point Z(t) = Zx(t) as ε → 0 of Zx,y(t) = Xx,y(t/ε) is a solution of the averaged equation dZ¯(t) = B¯(Z¯(t)), Z¯(0) = x. (1.7) dt t It is known that the limit in (1.3) is uniform in y if and only if the flow Fx on M is uniquely ergodic, i.e. it possesses a unique invariant measure, which occurs rather rarely. If, in addition, the uniform convergence in x is required, as well, we arrive at very few specific models of uniquely ergodic families. Thus, the uniform convergence in (1.3) assumption is too restrictive and excludes many interesting cases. Probably, the first relatively general result on fully coupled averaging is due AVERAGING PRINCIPLE 1189 to Anosov [1] (see also [22] and [21]). Relying on the Liouville theorem he showed t 1 that if each flow Fx preserves a probability measure µx on M having a C dependent on x density with respect to the Riemannian volume m on M then for any δ > 0,
ε ¯ ε mes{(x, y) : sup |Xx,y(t) − Xx(t)| > δ} → 0 as ε → 0, (1.8) 0≤t≤T/ε where mes is the product of m and the Lebesgue measure in a relatively compact domain X ⊂ Rd. An improvement of (1.8) to (1.6) is not possible in a general fully coupled situation (see Appendix in [21]). Observe that Anosov’s theorem does not cover other situations where the aver- aging principle is known to work and [20] give necessary and sufficient conditions d which ensure that for given probability measures ν on R and µx on M, Z Z ε ¯ ε sup |Xx,y(t) − Xx(t)|dµx(y)dν(x) → 0 as ε → 0. (1.9) X M 0≤t≤T/ε
It was shown in [20] that these conditions hold true when fast motions are hyper- bolic flows. More specifically, [20] considers the situation where b(x, y) is C2 in x and y and for each x in a closure of a relatively compact domain X the flow t Fx is Anosov or, more generally, Axiom A in a neighborhood of an attractor Λx. SRB t Let µx R be the Sinai-Ruelle-Bowen (SRB) invariant measure of Fx on Λ and set ¯ SRB ¯ B(x) = B(x, y)dµx (y). It is known (see [9]) that the vector field B(x) is Lip- schitz continuous in x, and so the averaged equations (1.5) and (1.7) (with right hand side independent of y) have unique solutions X¯ ε(t) and Z¯(t) = X¯ ε(t/ε). Still, SRB in general, the measures µx are singular with respect to the Riemannian volume on M, and so the method of [1] cannot be applied here. The general conditions from [20] were verified there for this situation relying on some large deviations argu- SRB ments which yielded (1.9) both with µx = µx and with µx being the Riemannian volume on M. Moreover,it turned out that in this situation the convergence to zero in (1.8) is exponentially fast in 1/ε. The appropriate large deviations machinery works only in the uniformly hyperbolic situation and in the present paper we give another simpler proof of the averaging principle in the form (1.9) for the above situation which should work also for some partially hyperbolic and nonuniformly hyperbolic fast motions. This simpler proof does not give however the exponentially fast convergence in (1.8). The corresponding problems for the averaging in the discrete time case with difference equations in place of (1.1) were studied in ([19]). Namely, consider se- ε ε ε ε quences of functions X (n) = Xx,y(n) and Y (n) = Yx,y(n), n = 0, 1, 2, ..., given by recurrence relations
Xε(n + 1) − Xε(n) = εΨ(Xε(n),Y ε(n)),Xε(0) = x, (1.10) Y ε(n + 1) = F (Xε(n),Y ε(n)),Y ε(0) = y, where Ψ is a smooth vector function and Fx = F (x, ·): M → M is a smooth map (a diffeomorphism or an endomorphism). The evolution of the system is described now by the map Φε(x, y) = (x+εΨ(x, y),Fxy) which can be considered as a perturbation of the map Φ0(x, y) = (x, Fxy) describing an idealized model with parameters x being integrals of motion. As usual in dynamical systems, it is quite useful to consider the discrete time setup whenever possible since it provides a richer source of examples than the continuous time (flow) case and it may better clarify the 1190 YURI KIFER situation. Now, if the limit NX−1 ¯ ¯ 1 n Ψ(x) = Ψy(x) = lim Ψ(x, Fx y) (1.11) N→∞ N n=0 exists, it is the same for ”many” y’s and Ψ(¯ x) is Lipschitz continuous then we can speak again about the averaged equation dX¯ ε(t) = εΨ(¯ X¯ ε(t)), X¯ ε(0) = x (1.12) dt ε ¯ ε and study the approximation of Xx,y(n) by Xx(n) for n ∈ [0, T/ε]. It was shown in [19] that the results similar to the continuous time case holds true for this discrete time setup, as well. A general sufficient condition was introduced there which d ensures that for given probability measures ν on R and µx on M, Z Z ε ¯ ε max |Xx,y(n) − Xx(n)|dµx(y)dν(x) → 0 as ε → 0. (1.13) X M 0≤n≤T/ε Relying on some large deviations arguments it was shown in [19] that the sufficient 2 condition in question holds true when fast motions Fx are C dependent on x Anosov or Axiom A diffeomorphisms (in the latter case considered in a neighborhood of an attractor) or expanding endomorphisms, and so the averaging principle in the form (1.13) holds true in this case. Since appropriate large deviations bounds can only be derived in the uniformly hyperbolic setup we give in this paper a simpler proof of the required sufficient condition which should work also for some partially and nonuniformly hyperbolic fast motions. This should enable us to extend the averaging principle in the form of (1.3) to other situations, in particular, we have in mind group extensions of Axiom A systems and examples considered in [2], [8], [10], [11] and [24].
2. Preliminaries and Main Result. We consider a system of differential equa- tions (1.1) on the product X¯ × M where X ⊂ Rd is an open set, X¯ is its closure and M is a compact C2 Riemannian manifold. We assume that there exists L > 0 such that for all x, z ∈ X¯ and y, v ∈ M,
kB(x, y) − B(z, v)k + kb(x, y) − b(z, v)k ≤ L(|x − z| + dM(y, v)) (2.1) and kB(x, y)k + kb(x, y)k ≤ L where dM is the distance on M. Together with (1.1) we consider also the equation (1.5) on X¯ with coefficients B¯ such that for some L¯ > 0 and all x, z ∈ X¯, kB¯(x) − B¯(z)k ≤ L¯|x − z| and kB¯(x)k ≤ L.¯ (2.2) The Lipschitz continuity conditions (2.1) and (2.2) ensure existence and uniqueness of solutions of (1.1) and (1.5), respectively. If B¯ is defined by (1.4) then (2.2) is equivalent to the existence of L˜ > 0 such that for all x, z ∈ X , Z ¯ ¯ ¯ B(x, y)d(µx − µz)(y)¯ ≤ L˜|x − z| (2.3) M ε which is a condition of regular dependence of µx on x. Set Xt = {x ∈ X : Xx,y(s) ∈ ¯ ε X , Xx(s) ∈ X for all y ∈ M and s ∈ [0, t/ε]}. It is clear that Xt is an open set and AVERAGING PRINCIPLE 1191
¯ by (2.1) and (2.2) it follows that Xt ⊃ {x ∈ X : infz∈X / |z − x| > 2t max(L, L)}. Introduce Z ¯ t ¯ ¯1 u ¯ ¯ E(t, δ) = {(x, y) ∈ Xt × M : B(x, Fx y)du − B(x) > δ}. t 0 The following result was proved in [20]. Theorem 2.1. Suppose that (2.1) and (2.2) hold true and let µ be a probability measure on X ×M. Assume that there exists an integer valued function n = n(ε) → T 1 ∞ as ε → 0 such that t(ε) = εn(ε) = o(log ε ) and for any δ > 0,
¡ −jt(ε) ¢ lim max µ (XT × M) ∩ Φε E(t(ε), δ) = 0. (2.4) ε→0 0≤j Then Z Z ε ¯ ε lim sup |Xx,y(t) − Xx(t)|dµ(x, y) = 0. (2.5) ε→0 XT M 0≤t≤T/ε Theorem 2.1 gives conditions for convergence in average in the averaging princi- ple. In view of resonances (see, for instance, [22]) it is impossible for many inter- esting examples to ensure (1.6) for all x ∈ X and µx-almost all (a.a) y where µx is a reasonable family of probability measures on M. Moreover, Neistadt’s example described in Appendix to [21] shows that the convergence (1.6) may even not hold true for all initial condition (from a large domain). In what follows we will verify (2.4) in the case of fast motions being slowly changing Axiom A flows. First, we recall basic definitions (see [7] and [16]). Let F t be a C2 flow on a compact Riemannian manifold M given by a differential equation dF ty = b(F ty),F 0y = y. dt A compact F t−invariant set Λ ⊂ M is called hyperbolic if there exists κ > 0 and s 0 u s 0 u the splitting TΛM = Γ ⊕ Γ ⊕ Γ into the continuous subbundles Γ , Γ , Γ of the tangent bundle T M restricted to Λ, the splitting is invariant with respect to the differential DF t of F t, Γ0 is the one dimensional subbundle generated by the vector s u field b, and there is t0 > 0 such that for all ξ ∈ Γ , η ∈ Γ , and t ≥ t0, kDF tξk ≤ e−κtkξk and kDF −tηk ≤ e−κtkηk. (2.6) t A hyperbolic set Λ is said to be basic hyperbolic if the periodic orbits of F |Λ are t dense in Λ,F |Λ is topologically transitive, and there exists an open set U ⊃ Λ with t Λ = ∩−∞ t0 t F U¯ ⊂ U and ∩t>0 F U = Λ where U¯ denotes the closure of U. If Λ = M then F t is called an Anosov flow. Assumption 2.2. The family b(x, ·) in (1.2) consists of C2 vector fields on an n−dimensional Riemannian manifold M with uniform C2 dependence on the pa- rameter x belonging to a relatively compact connected open set X and depending ¯ t ¯ continuously on x in its closure X . Each flow Fx, x ∈ X on M given by dF ty x = b(x, F ty),F 0y = y (2.7) dt x x 1192 YURI KIFER s 0 u possesses a basic hyperbolic attractor Λx with a splitting TΛx M = Γx ⊕ Γx ⊕ Γx satisfying (2.6) with the same κ > 0 and there exists an open set W ⊂ M and t0 > 0 such that t ¯ t ¯ Λx ⊂ W,FxW ⊂ W ∀t ≥ t0, and ∩t>0 FxW = Λx ∀x ∈ X . (2.8) u t u u t Let Jx (t, y) be the Jacobian of the linear map DFx(y):Γx(y) → Γx(Fxy) with respect to the Riemannian inner products and set dJ u(t, y)¯ ϕu(y) = − x ¯ . (2.9) x dt t=0 u The function ϕx(y) is known to be H¨oldercontinuous in y, since the subbundles u u 1 Γx are H¨oldercontinuous (see [7] and [16]), and ϕx(y) is C in x (see [9]). Let W t s ¯ t satisfies (2.8) and set Wx = {y ∈ W : Fx y ∈ W ∀s ∈ [0, t]}. A set E ⊂ Wx is s s called (δ, t)−separated for the flow Fx if y, z ∈ E, y 6= z imply d(Fx y, Fx z) > δ for some s ∈ [0, t], where d(·, ·) is the distance function on M. For each continuous R P t s t function ψ on W set Px(ψ, δ, t) = sup{ y∈E exp 0 ψ(Fx y)ds : E ⊂ Wx is (δ, t) − t separated for Fx},Px(ψ, δ, t) = 0 if Wx = ∅, and 1 Px(ψ, δ) = lim sup log Px(ψ, δ, t). t→∞ t The latter is monotone in δ, and so the limit Px(ψ) = lim Px(ψ, δ) δ→0 t exists and it is called the topological pressure of ψ for the flow Fx. Let Mx denotes t the space of Fx−invariant probability measures on Λx then (see, for instance, [16]) the following variational principle Z 1 Px(ψ) = sup ( ψdµ + hµ(Fx )) (2.10) µ∈Mx 1 1 holds true where hµ(Fx ) is the Kolmogorov–Sinai entropy of the time-one map Fx with respect to µ. If q is a H¨oldercontinuous function on Λx then there exists a t q u unique Fx−invariant measure µx on Λx, called the equilibrium state for ϕx +q, such that Z u u q q 1 Px(ϕx + q) = (ϕx + q)dµx + hµx (Fx ). (2.11) 0 SRB We denote µx by µx since it is usually called the Sinai–Ruelle– Bowen (SRB) measure. It has another important property (which extends to the more general nonuniformly hyperbolic setup) that its conditional measures on unstable manifolds (disintegrations with respect to the unstable foliation) are equivalent to the induced u Riemannian volume there. Since Λx are attractors we have that Px(ϕx) = 0 (see [7]). We replace now the condition (2.1) by the following stronger Assumption 2.3. There exist L > 0 such that for all x ∈ X¯, y ∈ M, kB(x, y)kC1(X¯ ×M) + kb(x, y)kC2(X¯ ×M) ≤ L (2.12) i ¯ where k · kCi(X¯ ×M) is the C norm of the corresponding vector fields on X × M Set Z ¯ SRB B(x) = B(x, y)dµx (y) (2.13) then under Assumption 2.3 B¯ is C1 in x (see [9]), and so (2.2) is automatically satisfied. AVERAGING PRINCIPLE 1193 Theorem 2.4. Suppose that Assumptions 2.2 and 2.3 hold true. Define B¯ by (2.13) and let µ be the product of a probability measure ν with support in XT and the normalized Riemannian volume mW on W. Then (2.4) is satisfied for some t(ε) → ∞ and n(ε) → ∞ as ε → 0 with εt(ε)n(ε) = T , and so (2.5) holds true. The result remains true if in place of the above we take µ defined by dµ(x, y) = SRB dν(x)dµx (y). It is still not clear whether in the setup of Theorem 2.4 it is possible to derive the convergence (1.6) for all (or for Lebesgue almost all) x ∈ XT and for mW -almost all y ∈ W and not just convergence in average (2.5). Next, we will formulate the corresponding results in the discrete time setup. We consider the system (1.10) on the product X¯ × M where X ⊂ Rd is an open set, X¯ is its closure and M is a compact C2 Riemannian manifold and assume that there exists L > 0 such that for all ε ≥ 0, x, z ∈ X¯ and y, v ∈ M, |Ψ(x, y) − Ψ(z, v)| + dM(Fxy, Fzv) ≤ L(|x − z| + dM(y, v)) (2.14) and |Ψ(x, y)| ≤ L d where Fx : M → M for each x ∈ R is a Lipschitz map and dM is the Riemannian metric on M. Assume also that for all x, z ∈ X , Z |Ψ(¯ x) − Ψ(x, y)dµz(y)| ≤ L¯|x − z| (2.15) M R ¯ ¯ ε for some L where Ψ(x) = M Ψ(x, y)dµx(y). Set Xt = {x ∈ X : Xx,y(k) ∈ ¯ ε ¯ ε X , Xx(k) ∈ X for all y ∈ M and k ∈ [0, t/ε]} where Xx(t) is the solution of (1.12). It is clear that Xt is an open set and by (2.14) and (2.15) it follows that Xt ⊃ {x ∈ ¯ X : infz∈X / |z − x| > 2t max(L, L)}. For each n ∈ N and δ > 0 set 1 kX−1 E(k, δ) = {(x, y) ∈ X × M : | Ψ(x, F my) − Ψ(¯ x)| > δ}. k k x m=0 The following result follows from [19]. Theorem 2.5. Suppose that (2.14) and (2.15) hold true and£ there exists an integer T 1 valued function n = n(ε) → ∞ as ε → ∞ such that `(ε) = εn(ε) ] = o(log ε ) and for any δ > 0, ¡ −j`(ε) ¢ lim max µ (XT × M) ∩ Φε E(`(ε), δ) = 0 (2.16) ε→0 0≤j 3. Dynamics of Perturbations. Any vector ξ ∈ T (Rd × M) = Rd ⊕ T M can be uniquely written as ξ = ξX + ξW where ξX ∈ T Rd and ξW ∈ T M and it has the Riemannian norm |kξ|k = |ξX | + kξW k where | · | is the usual Euclidean norm on Rd and k · k is the Riemannian norm on M. The corresponding metrics on M and d on R × M will be denoted by dM and d, respectively, so that if z1 = (x1, w1), z2 = d (x2, w2) ∈ R × M then d(z1, z2) = |x1 − x2| + dM(w1, w2). It is known (see [25]) s 0 u that the hyperbolic splitting TΛx M = Γx ⊕ Γx ⊕ Γx over Λx can be continuously s 0 u extended to the splitting TW M = Γx ⊕ Γx ⊕ Γx over W which is forward invariant s with respect to DFx and satisfies exponential estimates with a uniform in x ∈ X positive exponent which we denote again by κ > 0, i.e. we assume now that t −κt −t −κt kDFxξk ≤ e kξk and kDFx ηk ≤ e kηk (3.1) s u t provided ξ ∈ Γx(w), η ∈ Γx(Fxw), t ≥ t0, and w ∈ W. Moreover, by [9] (see s u also [26]) we can choose these extensions so that Γx(w) and Γx(w) will be H¨older continuous in w and C1 in x in the corresponding Grassmann bundle. Actually, since W is contained in the basin of each attractor Λx, any point w ∈ W belongs AVERAGING PRINCIPLE 1195 s to the stable manifold Wx (v) of some point v ∈ Λx (see [7]), and so we choose s s s naturally Γx(w) to be the tangent space to Wx (v) at w ∈ Wx (v). Now each vector ξ ∈ Tx,w(X × W) = TxX ⊕ TwW can be represented uniquely in the form ξ = X s 0 u X s s 0 0 u u ξ + ξ + ξ + ξ with ξ ∈ TxX , ξ ∈ Γx(w), ξ ∈ Γx(w) and ξ ∈ Γx(w). We denote also ξ0s = ξs + ξ0 and ξ0u = ξu + ξ0. For each small ε, α > 0 set Cu(ε, α) = 0s −2 u X −1 u u {ξ ∈ T (X × W): kξ k ≤ εα kξ k and kξ k ≤ εα kξ k} and Cx,w(ε, α) = u u u C (ε, α)∩Tx,w(X ×W) which are cones around Γ and Γx(w), respectively. Similarly, we define Cs(ε, α) = {ξ ∈ T (X × W): kξ0uk ≤ εα−2kξsk and kξX k ≤ εα−1kξsk} s s s s and Cx,w(ε, α) = C (ε, α) ∩ Tx,w(X × W) which are cones around Γ and Γx(w), τ respectively. Put (X × W)t = {(x, w):Φε (x, w) ∈ (X × W) ∀τ ∈ [0, t]}, where, t recall, Φε is the flow determined by the equations (1.1). The following result was proved in [20]. Lemma 3.1. There exist α0, ε(α), t1 > 0 such that if z = (x, y) ∈ (X × W)t and t ≥ t1, α ≤ α0, ε ≤ ε(α) then t u u s −t s DzΦ C (ε, α) ⊂ C t (ε, α), C (ε, α) ⊃ DzΦ C t (ε, α), (3.2) ε z Φεz z ε Φεz u s and for any ξ ∈ C (ε, α), η ∈ C t , z Φεz 1 1 t 2 κt −t 2 κt |kDzΦεξ|k ≥ e |kξ|k, |kDzΦε η|k ≥ e |kη|k. (3.3) d Ξ For any linear subspace Ξ of Tz(R × M) denote by Jε (t, z) absolute value of t t the Jacobian of the linear map DzΦε :Ξ → DzΦεΞ with respect to inner products induced by the Riemannian metric. For each z = (x, y) ∈ Rd × M set also Z t u ¡ u ε ¢ J (t, z) = exp − ϕ ε (Y (s))ds . (3.4) ε Xx,y (s) x,y 0 u Let nu be the dimension of Γx(w) which does not depend on x and w by continuity considerations. If Ξ is an nu−dimensional subspace of Tz(X × W), z = (x, y), and u Ξ ⊂ Cx,y(ε, α) then it follows easily from Assumptions 2.2, 2.3 and Lemma 3.1 that there exists a constant C1 > 0 independent of z ∈ X × W and a small ε such that for any t ≥ 0, t Ξ u −1 t (1 − C1ε) ≤ Jε (t, z)(Jε (t, z)) ≤ (1 + C1ε) . (3.5) d u Denote by U(z, ρ) the ball in R × M centered at z and let Dε (z, α, ρ, C) be 1 the set of all C embedded nu−dimensional closed discs D ⊂ X × W such that u 2 z ∈ D, TD ⊂ C (ε, α) and if v ∈ ∂D then Cρ ≤ dD(v, z) ≤ C ρ where TD is the tangent bundle over D, ∂ denotes the boundary, and dD is the interior metric on u 2 2 D. If D ∈ Dε (z, α, ρ, C) then, clearly, D ⊂ U(z, C ρ) and if ε/α and ρ are small u enough then d(v, z) ≥ ρ for any v ∈ ∂D. Let D ∈ Dε (z, α, ρ, C) and z = (x, y) ∈ ε s s D ⊂ X × W. Set UD(t, z, γ) = {z˜ ∈ D : dD(Φεz, Φεz˜) ≤ γ ∀s ∈ [0, t]} and let π1 : X × W → X and π2 : X × W → W be natural projections on the first and second factors, respectively. The proof of the following result can be found in [20]. Lemma 3.2. Let ε, α, t, (x, y) be as in Lemma 3.1 and T > 0. There exist u ρ0, c, cρ,T , C > 0 such that if ρ ≤ ρ0,D ∈ Dε ((x, y), α, ρ, C), z ∈ D, Vs,t(z) = s ε ΦεUD(t, z, Cρ), Vt(z) = Vt,t(z), V = V0,t(z) ⊂ D = ∅ and t ≥ 0 then 1 1 s s −1 − 2 κ(t−s) t t −1 − 2 κ(t−s) (i) dVs,t(z)(Φεv, Φεz) ≤ c e dVt(z)(Φεv, Φεz) ≤ c Cρe for any v ∈ V and s ∈ [0, t], where dU is the interior distance√ on U; u u t (ii) TVs,t(z) ⊂ C (ε, α) and Vt(z) ∈ Dε (Φεz, α, ρ, C); (iii) For all v ∈ V and 0 ≤ s ≤ t, 1 s s −1 −1 −1 −2 −1 − 2 κ(t−s) |π1Φεv − π1Φεz| ≤ Cc ρεα (1 − εα − εα ) e . 1196 YURI KIFER u −1 (iv) cρ,T ≤ mD(V )Jε (t, z) ≤ cρ,T provided t ≤ T/ε, where mD is the induced Riemannian volume on D. Next, we consider the discrete time setup. As before we consider the represen- tation of any vector ξ ∈ T (Rd × M) = Rd ⊕ T M in the form ξ = ξX + ξW where ξX ∈ T Rd and ξW ∈ T M and we use its Riemannian norm |kξ|k = |ξX | + kξW k where | · | is the usual Euclidean norm on Rd and k · k is the Riemannian norm on M. The corresponding metrics on M and on Rd × M will be denoted, again, by dM and d, respectively. It is known (see [14] and [25]) that the hyperbolic s u splitting TΛx M = Γx ⊕ Γx over Λx can be continuously extended to the splitting s u s TW M = Γx ⊕ Γx over W which is forward invariant with respect to DFx and sat- isfies exponential estimates with a uniform in x ∈ X positive exponent which we denote again by κ > 0, i.e. we assume now that n −κn −n −κn kDFx ξk ≤ e kξk and kDFx ηk ≤ e kηk (3.6) s u n provided ξ ∈ Γx(w), η ∈ Γx(Fx w), n ≥ n0, and w ∈ W. Moreover, by [9] (see s u also [26]) we can choose these extensions so that Γx(w) and Γx(w) will be H¨older continuous in w and C1 in x in the corresponding Grassmann bundle. Actually, since W is contained in the basin of each attractor Λx, any point w ∈ W be- s longs to the stable manifold Wx (v) of some point v ∈ Λx (see [7]), and so we s s choose naturally Γx(w) to be the tangent space to Wx (v) at w. Now each vec- tor ξ ∈ Tx,w(X × W) = TxX ⊕ TwW can be represented uniquely in the form X s u X s s u u ξ = ξ + ξ + ξ with ξ ∈ TxX , ξ ∈ Γx(w), and ξ ∈ Γx(w). For each small ε, α > 0 set Cu(ε, α) = {ξ ∈ T (X × W): kξsk ≤ εα−2kξuk and kξX k ≤ −1 u u u u εα kξ k} and Cx,w(ε, α) = C (ε, α) ∩ Tx,w(X × W) which are cones around Γ u s u and Γx(w), respectively. Similarly, we define C (ε, α) = {ξ ∈ T (X × W): kξ k ≤ −2 s X −1 s s s εα kξ k and kξ k ≤ εα kξ k} and Cx,w(ε, α) = C (ε, α) ∩ Tx,w(X × W) which s s k are cones around Γ and Γx(w), respectively. Put (X × W)n = {(x, w):Φε (x, w) ∈ (X × W) ∀k = 0, 1, ..., n}, where, recall, Φε(x, v) = (x + εΨ(x, y),Fxv). Now, in this notations Lemma 3.1 holds true again by a trivial simplification of the proof in [20] (see also [19]). d Ξ For any linear subspace Ξ of Tz(R × M) denote by Jε (n, z) absolute value of n n the Jacobian of the linear map DzΦε :Ξ → DzΦε Ξ with respect to inner products induced by the Riemannian metric. For each z = (x, y) ∈ Rd × M set also nX−1 u ¡ u ε ¢ J (n, z) = exp − ϕ ε (Y (k)) . (3.7) ε Xx,y (k) x,y k=0 u Let nu be the dimension of Γx(w) which does not depend on x and w by continuity considerations. If Ξ is an nu−dimensional subspace of Tz(X × W), z = (x, y), and u Ξ ⊂ Cx,y(ε, α) then it follows easily from our assumptions and Lemma 3.1 that there exists a constant C1 > 0 independent of z ∈ X × W and a small ε such that for any t = n ≥ 0 the inequality (3.5) holds true. Furthermore, in the corresponding notations Lemma 3.2 remains valid for the discrete time setup, as well, with the proof which is again a slight simplification of the one in [20] (see also [19]). The corresponding results for C2 expanding endomorphisms follow easily by another simplification, essentially, just disregarding stable subbundles and foliations. 4. Proofs and Concluding Remarks. For each y ∈ Λx and γ > 0 small enough s t t u set Wx (y, γ) = {y˜ ∈ W : dM(Fxy, Fxy˜) ≤ γ ∀t ≥ 0} and Wx (y, γ) = {y˜ ∈ W : t t dM(Fxy, Fxy˜) ≤ γ ∀t ≤ 0} which are local stable and unstable manifolds for Fx at AVERAGING PRINCIPLE 1197 y. According to [14] and [25] these families can be included into continuous families 1 s u of ns and nu−dimensional stable and unstable C discs Wx (y, γ) and Wx (y, γ), re- s s u spectively, defined for all y ∈ W and such that Wx (y, γ) is tangent to Γx, Wx (y, γ) u t s s t u −t u t is tangent to Γx, FxWx (y, γ) ⊂ Wx (Fxy, γ), and Wx (y, γ) ⊃ Fx Wx (Fxy, γ). Ac- s tually, as we noted it above if y ∈ W then y belongs to a stable manifold Wx (˜y) of s s somey ˜ ∈ Λx and we choose Wx (y, γ) to be the subset of Wx (˜y). x r r For any set V ⊂ W put UV (t, y, ζ) = {v ∈ V : d(Fx y, Fx v) ≤ ζ ∀r ∈ [0, t]} and x x U (t, y, ζ) = UW (t, y, ζ). Recall, that a finite set E ⊂ W is called (ζ, t)−separated t x for the flow Fx if y, y˜ ∈ E, y 6=y ˜ impliesy ˜ 6∈ U (t, y, ζ). Observe that if E is a (ζ, t)−separated subset then UΛx (t, y, ζ/2), y ∈ E are disjoint sets. A finite set E ⊂ D will be called (s, γ, ε, D)-separated if vi, vj ∈ E, vi 6= vj imply that vi 6∈ ε UD(s, vj, γ). Let ν be a probability measure on X with G =suppν ⊂ XT . For small ρ and large C so that Cρ is still small we consider for each x ∈ G and y ∈ W the balls 0 u Dx(y) and Dx(y) on Wx (y, γ) centered at y and having radii Cρ and 2Cρ, respec- u 0 u tively, in the interior metric on Wx (y, γ). Then Dx(y),Dx(y) ∈ Dε ((x, y), α, ρ, C). Choose a maximal (t, Cρ, ε, Dx(y))-separated set E on Dx(y) (i.e. a set which cannot stay (t, Cρ, ε, Dx(y))-separated if we add to it points from Dx(y)). Then ∪ U ε (t, v, Cρ) ⊃ D (y) and by Lemma 3.2(i), U ε (t, v, Cρ) ⊂ D0 (y) for v∈E Dx(y) x Dx(y) x all x ∈ E provided t is large enough. Moreover, U ε (t, v, Cρ/2) are disjoint for Dx(y) different v ∈ E. For each v ∈ E consider V (v) = Φt U (t, v, Cρ) which by Lemma 3.2(ii) √ t ε Dx(y) u t belongs to Dε (vt, α, ρ, C) with vt = Φεv. By Lemma 3.2(iii), sup |π1w − π1vt| ≤ C2ε (4.1) w∈Vt(v) where C2 > 0 and other constants Ci below are independent of v ∈ Dx(y), t ≤ t/ε and ε > 0 but may depend on ρ. Choose τ(ε) = (log 1/ε)β for some β ∈ (0, 1) then by(4.1) for any w ∈ Vt(v), u u √ sup dM(Fπ1wπ2w, Fπ1vt π2w) ≤ ε (4.2) 0≤u≤τ(ε) provided ε is small enough, and so by (2.12) for any w ∈ Vt(v), Z τ(ε) 1 u u √ |B(π1w, Fπ1wπ2w) − B(π1vt,Fπ1vt π2w)|du ≤ 2L ε. (4.3) τ(ε) 0 By an appropriate version of the inverse function theorem (see, for instance, [5], V Ch.2 and [15], Ch.1,3) it follows that the restriction π2 to Vt(v) of the map π2 : Vt(v) → V˜ = π2(Vt(v)) is a diffeomorphism on the interior of Vt(v). Moreover, it is ˜ ˜ ˜ easy to see that for any w ∈ V the tangent space {0} × Tπ2vt V = Tvt ({π1vt} × V ) is u contained in Cvt (ε, α), and so its Grassmannian distance from Tvt Vt(v) is of order V V ˜ ε. We conclude from here that the Jacobian Jacwπ2 of π2 at any w ∈ V satisfies V 1 − C3ε < |Jacwπ2 | < 1 + C3ε. (4.4) for some C3 > 0 and all small ε. If ρ is sufficiently small then V˜ is contained in one coordinate chart Uvt of M centered at vt. Clearly, the Grassmannian distance of ˜ u u Tvt ({π1vt} × V ) from Tvt ({π1vt} × Wπ1vt (π2vt, γ)) ⊂ Cvt (ε, α) is of order ε, as well, ˜ u provided γ is not too small. In the coordinate chart Uvt project V to Wπ1vt (vt, γ) s 0 along the family of (ns +1)-dimensional planes parallel to Γπ1vt ⊕Γπ1vt (see Section 1198 YURI KIFER ˜ ˜ ˜ u 3) which yields a diffeomorphism ψ : intV → W = ψ(intV ) ⊂ Wπ1vt (vt, γ). It follows from above that for any w ∈ V˜ , dM(w, ψw) ≤ C4ε and 1 − C4ε < |Jacwψ| < 1 + C4ε (4.5) for some C4 > 0 and all ε small enough. By the choice of τ(ε) for all w ∈ V˜ , θ θ √ sup dM(Fπ1vt w, Fπ1vt ψw) ≤ ε 0≤θ≤τ(ε) provided ε is small enough, and so by (2.12), Z τ(ε) 1 θ θ √ |B(π1vt,Fπ1vt w) − B(π1vt,Fπ1vt ψw)|dθ ≤ L ε. (4.6) τ(ε) 0 Given δ > 0 we can choose r = r(δ) small enough so that if w ∈ W˜ = ψV˜ and w˜ ∈ U x(τ(ε), w, r) then Z τ(ε) ¯ ¯ 1 ¯ θ θ ¯ B(π1vt,Fπ1vt w) − B(π1vt,Fπ1vt w˜) dθ < δ/3. (4.7) τ(ε) 0 Employing the machinery of [9], in particular, the same symbolic dynamics for flows θ Fx with nearby x’s as in Sec. 5 of [9], together with uniform decay of correlations estimates for subshifts of finite type we obtain (cf. [17], p.1151) that there exists a constant C5 > 0 such that Z ¯ Z ¯ ¯ Θ ¡ ¢ ¯2 ¯ θ ¯ ¯ SRB sup ¯ B(x, Fx w) − B(x) dθ¯ dµx (w) ≤ C5Θ (4.8) x∈X 0 for all Θ ≥ 1. Hence, by Chebyshev’s inequality ¡ ¢ SRB 2 −1 µx E(τ(ε), γ) ≤ C5(γ τ(ε)) = ηγ (ε) → 0 as ε → 0. (4.9) ˜ ˜ θ Choose a maximal (r, τ(ε))-separated set E ⊂ W for the flow Fπ1vt . Then ¡ ¢ ˜ x ˜ ∪w∈E˜ W ∩ U (τ(ε), w, r) ⊃ W and (4.10) U x(τ(ε), w, r/2) ∩ U x(τ(ε), w,˜ r/2) = ∅ ∀w, w˜ ∈ E,˜ w 6=w. ˜ By [7] for any r > 0 there exists Cr > 0 such that ¡ ¢ ¡ ¢ ¡ ¢ ˜ x SRB x 2 SRB x mW˜ W ∩ U (τ(ε), w, r) ≤ Crµx U (τ(ε), w, r) ≤ Cr µx U (τ(ε), w, r/2) (4.11) ˜ where, recall, mW˜ is the induced Riemannian volume on W . Moreover, employing the technique from [9], in particular, using the same symbolic dynamics for flows θ u Fx with nearby x’s (see Sec. 5 in [9]) and the smooth dependence of ϕx on x we conclude that Cr > 0 above can be chosen the same for all x ∈ X . It follows from (4.9)–(4.11) that AVERAGING PRINCIPLE 1199 µ ¶ ¡ ¢ ˜ x ˜ mW˜ (E(τ(ε), δ) ∩ W ) ≤ mW˜ ∪w∈E˜∩E(τ(ε),2δ/3) U (τ(ε), w, r) ∩ W X ¡ ¢ x ˜ ≤ mW˜ U (τ(ε), w, r) ∩ W w∈E˜∩E(τ(ε),2δ/3) X ¡ SRB x ≤ Cr µx U (τ(ε), w, r)) w∈E˜∩E(τ(ε),2δ/3) (4.12) X ¡ 2 SRB x ≤ Cr µx U (τ(ε), w, r/2)) w∈E˜∩E(τ(ε),2δ/3) µ ¶ ¡ ¢ 2 SRB x ≤ Cr µx ∪w∈E˜ U (τ(ε), w, r/2) ∩ E(τ(ε), δ/3) 2 ≤ Cr ηδ/3(ε). This together with (4.3)–(4.7) give mVt(v)(Vt(v) ∩ E(τ(ε), δ)) ≤ C6η(ε) (4.13) where mVt(v) is the induced Riemannian volume on Vt(v) and, as before, C6 > 0 and other constants Ci here are independent of v ∈ Dx(y), w, w˜, t ≤ t/ε and ε > 0 but may depend on ρ and r. u By (3.4), Lemma 3.2(i) and the H¨oldercontinuity of ϕx(v) in both variables (and 1 even C in the x variable, see [9], [26] and [11]) we obtain that for any w, w˜ ∈ V0(v), −1 u ¡ u ¢−1 C7 ≤ Jε (t, w) Jε (t, w˜) ≤ C7 (4.14) for some C7 > 0. This together with (3.5) yield that for any w, w˜ ∈ V0(v), ¡ ¢−1 −1 Ξw Ξw˜ C8 ≤ Jε (t, w) Jε (t, w˜) ≤ C8 (4.15) u for some C8 > 0 where Ξw and Ξw˜ are tangent spaces to Wx (y, γ) at w andw ˜, T respectively. Now (4.9) together with (4.11) imply that uniformly in t ≤ ε − τ(ε), ¡ −t ¢¡ ¢−1 mDx(y) V0(v) ∩ Φε E(τ(ε), δ) mDx(y)(V0(v)) ≤ C9η(ε) (4.16) 0 for some C9 > 0. Since ∪v∈EV0(v) ⊃ Dx(y) and UDx(y)(t, v, Cρ/2) ⊂ Dx(y) are disjoint for different points v from E we conclude that ¡ −t ¢ mDx(y) Dx(y) ∩ Φε E(τ(ε), δ) ≤ C10η(ε) (4.17) for some C10 > 0. Set θ s Qx(y) = ∪{Dx(Fx y˜): |θ| ≤ Cρ, y˜ ∈ Wx (y, Cρ)} and Rx(y) = ∪|x˜−x|≤CρQx˜(y). We choose Cρ sufficiently small so that π1Rx(y) ⊂ SRB XT whenever x ∈ G. Recall that the conditional measures of µx on unstable manifolds are equivalent to the induced Riemannian volume there which follows from the two sided bounds of ratios ¡ ¢ ¡ ¢ −1 SRB x x C11 < µx U (θ, w, r) /mW U (θ, w, r) < C11, (4.18) with some C11 > 0 depending on r, the latter being a consequence of the two sided bounds for equilibrium states of Bowen’s balls from [12] and the volume lemma from [7] (observe that [7] proves only the left hand side of (4.18)). We can obtain the bounds (4.16) uniformly for all x ∈ X using, for instance, the same symbolic θ dynamics for flows Fx with nearby x’s as in Sec. 5 of [9]. Hence, we conclude 1200 YURI KIFER from (4.17) and the above construction that for both choices of the measure µ in Theorem 2.4, ¡ −t ¢ µ Rx(y) ∩ Φε E(τ(ε), δ) ≤ C12ηδ/3(ε) (4.19) for some C12 > 0. Covering G × W¯ by a finite number sets Rx(y) we obtain that T uniformly in t ≤ ε − τ(ε), ¡ −t ¢ µ (XT × M) ∩ Φε E(τ(ε), δ) ≤ C13ηδ/3(ε) (4.20) for some C13 > 0 which implies (2.4), and so (2.5) follows, completing the proof of Theorem 2.4. The proof of Theorem 2.6 proceeds in the same way as above by just disregarding the flow direction and by further disregarding the stable subbundle (and stable submanifolds) we derive the proof for the expanding endomorphisms case (observing that noninvertibility does not create additional problems here). The proof above involves two main ingredients. First, we employ some perturba- tion machinery described in Section 3. Secondly, we need what is naturally to call some bounded distortion arguments which ensure that the proportion of the volume of the set of irregular points on the disc Vt(v) (i.e. points from Vt(v) ∩ E(τ(ε), δ)) −t remains small after we map it back by Φε . 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