Center Manifolds Optimal Regularity for Nonuniformly Hyperbolic Dynamics 1

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1 São Paulo Journal of Mathematical Science 5, 1 (2011), 1 22 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 1 Lui Barreira and Claudia Vall Abtract. For ufficiently mall perturbation with continuou derivative, we how how to etablih the optimal regularity of invariant center manifold when the linear equation admit a nonuniform exponential trichotomy. We alo conider the general cae of exponential growth rate given by an arbitrary function. Thi include the uual exponential behavior a a very pecial cae. Our proof ue the fiber contraction principle to etablih the regularity property. We note that the argument alo applie to ufficiently mall linear perturbation, without further change. 1. Introduction It i widely recognized that center manifold theorem are powerful tool in the analyi of the behavior of nonhyperbolic dynamical ytem. For example, let u conider the dynamic generated by the differential equation u = A(t)u + f(t, u), (1) and let u aume that f(t, 0) = 0 for every t. One can ak wether the behavior of the olution of equation (1) in a neighborhood of zero omehow imitate the behavior of thoe of the linear equation u = A(t)u. (2) Thi i certainly the cae when equation (2) admit an exponential dichotomy. Indeed, by the Grobman Hartman theorem, locally the two dynamic are topologically conjugate. On the other hand, when equation (2) ha ome elliptic direction one can till etablih the exitence of invariant center manifold that are tangent to the ubpace generated by thee direction. However, the ituation i not o imple anymore. Namely, the Key word: nonuniform exponential trichotomie, center manifold. 1 Partially upported by FCT through CAMGSD, Libon. 1

2 2 Lui Barreira and Claudia Vall behavior on a center manifold ubtantially depend on the nonlinear perturbation, and in general the behavior of the dynamic on the manifold need not imitate the behavior on the ubpace. On the other hand, thi behavior eentially determine the tability of the ytem. We note that one i till able to how that along the hyperbolic direction locally the two dynamic are topologically conjugate, that i, one can partially linearize the nonlinear dynamic, namely along the hyperbolic direction. Accordingly, the tudy of center manifold play a crucial role in determining the tability of olution of a differential equation. Namely, when equation (2) ha no untable direction, all olution converge exponentially to the center manifold. Therefore, the tability of the ytem i completely determined by the behavior on the center manifold. Thu, one often conider a reduction to the center manifold, and determine the quantitative behavior on it. Thi ha alo the advantage of reducing the dimenion of the ytem. We refer the reader to the book [6] for detail and reference. In particular, uing normal form there i alo the poibility of an appropriate decription of the poible bifurcation. We note that ince one need to approximate the center manifold to ufficiently high order, it i alo important to dicu their moothne, and to undertand how to approximate them up to a given order. We concentrate in thi paper on the tudy of the moothne of invariant center manifold for nonautonomou differential equation. In it claical formulation, the center manifold theorem applie to flow for which the linear equation (2) admit a uniform exponential trichotomy. In particular, thi mean that the exponential etimate for the olution are aumed to be independent of the initial time. The tudy of center manifold can be traced back to the work of Pli [19] and Kelley [15]. A very detailed expoition in the cae of autonomou equation i given in [20], adapting reult in [22]. See alo [16, 21] for the cae of equation in infinite-dimenional pace. We refer the reader to [8, 9, 10, 20] for more detail and further reference. It i difficult to give an original reference for the firt publihed verion of a center manifold theorem in the nonautonomou cae. Certainly, the modification that are neceary to pa from autonomou to nonautonomou ytem are not ubtantial. In contrat, our main goal i to weaken the condition concerning the exitence of a uniform exponential trichotomy, to include a nonuniform exponential behavior. In thi cae the partial hyperbolicity can be poiled exponentially along each olution a the initial time change. We note that a principal motivation for weakening the aumption of uniform exponential behavior for equation (2) i that almot all trajectorie of a mooth dynamical ytem preerving a finite invariant meaure have a linear variational equation with a nonuniform exponential trichotomy (ee [3] for more detail and further reference).

3 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 3 A it i already mentioned above, we conider the general cae of nonuniform exponential behavior. In thi repect, our reult are alo a contribution to the theory of nonuniform hyperbolicity. We refer to [1] for a detailed expoition of the theory, which goe back to the landmark work of Oeledet [17] and particularly Pein [18]. Since then it became an important part of the general theory of dynamical ytem and a principal tool in the tudy of tochatic behavior. We note that the nonuniform hyperbolicity condition can be expreed in term of the Lyapunov exponent. Among the mot important propertie due to nonuniform hyperbolicity i the exitence of table and untable invariant manifold, and their abolute continuity property etablihed by Pein. The theory alo decribe the ergodic propertie of dynamical ytem with a finite invariant meaure abolutely continuou with repect to the volume, and it expree the Kolmogorov Sinai entropy in term of the Lyapunov exponent by Pein entropy formula. In another direction, combining the nonuniform hyperbolicity with the nontrivial recurrence guaranteed by the exitence of a finite invariant meaure, the fundamental work of Katok revealed a very rich and complicated orbit tructure, including an exponential growth rate for the number of periodic point meaured in term of the topological entropy, and the approximation of the entropy of an invariant meaure by uniformly hyperbolic horehoe. We refer to the book [1] for detail and reference. To the bet of our knowledge, the firt center manifold theorem on the exitence of mooth center manifold in the context of nonuniform exponential behavior wa obtained in [2], where we etablihed the C k regularity of the center manifold. We note however that thi i not an optimal reult ince we require the top C k derivative of the perturbation to be Lipchitz. Thi obervation i the main motivation for the preent work, and our main aim i preciely to obtain the optimal moothne of the center manifold. For implicity of the expoition we conider only the C 1 cae. The cae of higher moothne can alo be conidered although at the expene of rather involved computation, which eentially require a convenient framework in order to obtain the required etimate. Incidentally, in [2] our proof of the C k moothne of the manifold ue a ingle fixed point problem in an appropriate complete metric pace. We ue a reult in [13], going back to a lemma of Henry in [14], which preciely allow u to etablih the exitence and imultaneouly the regularity of the center manifold uing a ingle fixed point problem, intead of needing an additional fixed point problem for each higher-order derivative. Eentially, the reult ay that the cloed unit ball in the pace C k,δ of function of cla C k with Hölder continuou kth derivative with Hölder exponent δ i cloed with repect to the C 0 -topology. Thi allow u to conider contraction map olely uing the upremum norm intead of needing any norm involving alo the derivative. See [11] for a related approach in the particular cae of uniform exponential

4 4 Lui Barreira and Claudia Vall behavior. In order to obtain the required etimate, we need harp bound for the derivative of the central component of the olution, and for the derivative of the vector field along a given graph. For thi we ue a multivariate verion of the Faà di Bruno formula in [12] for the derivative of a compoition. Here we ue an entirely different approach, which allow u to obtain the optimal moothne of the center manifold. Namely, we ue the fiber contraction principle together with an elaboration of an argument ketched in [7] (although now in the nonuniform etting and for arbitrary growth rate) to etablih the continuity of fiber contraction. Thi lat tep turn out to be the main technical difficulty of the proof. We briefly recall the fiber contraction principle. Given metric pace X and Y = (Y, d Y ), we ay that a tranformation S : X Y X Y of the form S(x, y) = (T (x), A(x, y)) for ome function T : X X and A: X Y Y i a fiber contraction if there exit λ (0, 1) uch that d Y (A(x, y), A(x, ȳ)) λ d Y (y, ȳ) for every x X and y, ȳ Y. We alo ay that a fixed point x 0 X of T i attracting if for every x X we have T n (x) x 0 when n. Propoition 1 (Fiber contraction principle). If S i a continuou fiber contraction, x 0 X i an attracting fixed point of T, and y 0 Y i a fixed point of A(x 0, ), then (x 0, y 0 ) i an attracting fixed point of S. We alo follow partially argument in [5] although now for arbitrary growth rate. Namely, we conider linear equation a in (1) that may exhibit central, table and untable behavior with repect to arbitrary growth rate e cρ(t) determined by an arbitrary function ρ. The uual exponential behavior i included a a very pecial cae when ρ(t) = t. Thee arbitrary growth rate include for example ituation in which all Lyapunov exponent of equation (1) are infinite (either + or ). Finally, we note that we have already etablihed the exitence of Lipchitz center manifold for arbitrary growth rate in [4], and o we imply refer to that paper in the Lipchitz part of proof. 2. Setup Let A: R M p be a C 1 function, where M p i the et of p p matrice. We conider the initial value problem u = A(t)u, u() = u (3) for each R and u X = R p. It unique olution i defined for every t R, and we write it in the form u(t) = T (t, )u, where T (t, ) i

5 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 5 the aociated linear evolution operator. Now we introduce the notion of exponential trichotomy, conidering both nonuniform exponential behavior, and growth rate given by an arbitrary function ρ. More preciely, given a trictly increaing differentiable function ρ: R R, we ay that equation (3) admit a ρ-nonuniform exponential trichotomy if there exit continuou function P, Q 1, Q 2 : R M p, uch that P (t), Q 1 (t) and Q 2 (t) are projection for each t R atifying and P (t) + Q 1 (t) + Q 2 (t) = Id, P (t)t (t, ) = T (t, )P (), Q i (t)t (t, ) = T (t, )Q i (), i = 1, 2 for every t, R, and there exit contant uch that for every t we have and for every t we have a < c c < b and ε, D > 0, (4) T (t, )P () De c(ρ(t) ρ())+ε ρ(), T (t, ) 1 Q 2 (t) De b(ρ(t) ρ())+ε ρ(t), T (t, )P () De c(ρ() ρ(t))+ε ρ(), T (t, ) 1 Q 1 (t) De a(ρ() ρ(t))+ε ρ(t). In thi cae we define center, table and untable ubpace for each R by E() = P ()X and F i () = Q i ()X, i = 1, 2. Setting ρ(t) = t we recover the notion of nonuniform exponential trichotomy (ee [3]). We preent an example taken from [4]. Example 1. Take β > δ > 0. For each t R \ {0} let δ ( co t 1 t in t ) t A(t) = 0 a(t) 0, 0 0 a(t) where a(t) = 1 2 t b( t ) + t b ( t ) δ 2 t in t δ t co t for ome C 1 function b: R + 0 R uch that b(t) lim = 0, b [1, ) = β, t 0 + t (5) (6)

6 6 Lui Barreira and Claudia Vall and b (t) > 0 for every t (0, 1). Setting A(0) = 0 we obtain a continuou function R t A(t). Then equation (3) admit a nonuniform exponential trichotomy for which the contant ε in (4) cannot be taken equal to zero. We alo conider a C 1 function f : R X X. We aume that f(t, 0) = 0 for every t R, and that there exit δ > 0 uch that f (t, u) u δρ (t)e 3ε ρ(t) (7) for every t R and u X. In addition, we aume that f(t, u) = 0 for every t R and u X with u c, for ome contant c > 0. Given R and u = (ξ, η 1, η 2 ) E() F 1 () F 2 (), we denote by (x(t), y 1 (t), y 2 (t)) = (x(t,, u ), y 1 (t,, u ), y 2 (t,, u )), with value in E(t) F 1 (t) F 2 (t), the unique olution of the initial value problem u = A(t)u + f(t, u), u() = u, (8) which i equivalent to x(t) = T (t, )ξ + y i (t) = T (t, )η i + P (t)t (t, )f(τ, x(τ), y 1 (τ), y 2 (τ)) dτ, Q i (t)t (t, )f(τ, x(τ), y 1 (τ), y 2 (τ)) dτ, i = 1, 2. One can how that each olution i defined for every t R. u(t) = 0 i a olution of equation (8). For each τ R we write Moreover, Ψ λ τ (, u ) = ( + τ, x( + τ,, u ), y 1 ( + τ,, u ), y 2 ( + τ,, u )). Thi i the flow generated by the autonomou equation t = 1, 3. Exitence of center manifold u = A(t)u + f(t, u). We formulate and prove our main reult in thi ection. It etablihe the exitence of C 1 center manifold for equation (8), or more preciely for the origin of equation (8). We emphaize that thi i the optimal regularity. The center manifold are obtained a graph over the center ubpace. We firt decribe the cla of function of which we conider the graph. Let alo X be the pace of continuou function φ = (φ 1, φ 2 ): { (, ξ) R X : ξ E() } X,

7 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 7 uch that φ(, 0) = 0, φ(, E()) F 1 () F 2 (), and φ(, ξ) φ(, ξ) ξ ξ (9) for every R and ξ, ξ E(). We can eaily verify that X i a complete metric pace with the ditance { } φ(, ξ) ψ(, ξ) d(φ, ψ) = up : R and ξ E() \ {0}. (10) ξ Given φ X, we define V φ = { (, ξ, φ(, ξ)) : (, ξ) R E() }. (11) The following i our main reult. Theorem 1. Let A and f be C 1 function. If the equation u = A(t)u admit a ρ-nonuniform exponential trichotomy with c b + ε < 0 and a c + ε < 0, f(t, 0) = f(t, u) = 0 for every t R and u X with u c, for ome contant c > 0, and condition (7) hold, then provided that δ i ufficiently mall there exit a unique φ X uch that Ψ τ (V φ ) = V φ for every τ R. (12) Moreover: 1. the function ξ φ(, ξ) i of cla C 1 for each R, and V φ i a C 1 manifold; 2. if ( f/ u)(t, 0) = 0 for every t R, then ( φ/ ξ)(, 0) = 0 and T (,0) V φ = R E() (13) for every R; 3. for each R and ξ, ξ E(), we have Ψ t (, ξ, φ(, ξ)) Ψ t (, ξ, φ(, ξ)) { 2De (c+2δd)(ρ(t) ρ())+ε ρ() ξ ξ, t, 2De ( c+2δd)(ρ() ρ(t))+ε ρ() ξ ξ, t. Proof. We note that property (12) i equivalent to the identitie x(t) = T (t, )ξ + φ i (t, x(t)) = T (t, )φ i,λ (, ξ) + P (t)t (t, τ)f(τ, x(τ), φ(τ, x(τ))) dτ, Q i (t)t (t, τ)f(τ, x(τ), φ(τ, x(τ))) dτ (14) (15)

8 8 Lui Barreira and Claudia Vall for every i = 1, 2, and t R. The following tatement etablihe the exitence of a olution of the firt equation in (15) for each given φ. Lemma 1. For each φ X and (, ξ) R E() there exit a unique continuou function x φ : { (, ξ) R X : ξ E() } X with x φ (, ξ) = ξ and x φ (t, ξ) E(t) for each t R, atifying the firt identity in (15) for every t R. Moreover, { 2De x φ (t, ξ) (c+2δd)(ρ(t) ρ())+ε ρ() ξ, t, 2De ( c+2δd)(ρ() ρ(t))+ε ρ() ξ, t. Lemma 1 follow from [4, Lemma 1 and 2] (etting φ = ψ and ξ = 0 in the econd lemma). Since Ψ t (, ξ, φ(, ξ)) = ( t, x φ (t, ξ), φ(t, x φ (t, ξ)) ), the inequalitie in (14) follow readily from Lemma 1 and (9). We have alo the following etimate. Lemma 2 ([4, Lemma 2]). For every φ, ψ X, R and ξ, ξ E(), we have x φ (t, ξ) x φ (t, ξ) + x φ (t, ξ) x ψ (t, ξ) De ( (c+2δd)(ρ(t) ρ())+ε ρ() ξ ξ ) e4δd(ρ(t) ρ()) ξ d(φ, ψ), t, De ( ( c+2δd)(ρ() ρ(t))+ε ρ() ξ ξ ) e4δd(ρ(t) ρ()) ξ d(φ, ψ), t. We notice that x φ i the unique olution of the equation x = A(t)x + P (t)f(t, x, φ(t, x)) (16) with initial condition x() = ξ. By Lemma 2 and the continuou dependence of the olution of a differential equation on the initial condition, the function (t, φ,, ξ) x φ (t, ξ) i continuou. Now we conider the two remaining equation in (15), and we rewrite them in a more convenient form. Lemma 3. For every ufficiently mall δ > 0, given φ X the following propertie are equivalent:

9 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 9 1. for every, t R, ξ E(), and i = 1, 2 we have φ i (t, x φ (t, ξ)) = T (t, )φ i (, ξ) + 2. for every R and ξ E() we have φ 1 (, ξ) = φ 2 (, ξ) = Q i (t)t (t, τ)f(τ, x φ (τ, ξ), φ(τ, x φ (τ, ξ))) dτ; Q 1 ()T (τ, ) 1 f(τ, x φ (τ, ξ), φ(τ, x φ (τ, ξ))) dτ, Q 2 ()T (τ, ) 1 f(τ, x φ (τ, ξ), φ(τ, x φ (τ, ξ))) dτ. (17) We can then how that there i a unique function φ atifying the two identitie in (17). Lemma 4 ([4, Lemma 4]). Provided that δ i ufficiently mall, there exit a unique φ X uch that (17) hold for every R and ξ E(). The function φ in Lemma 4 i the unique fixed point of a contraction operator T in the pace X. Namely, the operator i given by ( (T φ)(, ξ) = Q 1 ()T (τ, ) 1 f(τ, x φ (τ, ξ), φ(τ, x φ (τ, ξ))) dτ, ) (18) Q 2 ()T (τ, ) 1 f(τ, x φ (τ, ξ), φ(τ, x φ (τ, ξ))) dτ for each φ X. Now we etablih the C 1 regularity of the map ξ φ(, ξ) for each R. Thi i the mot delicate part of the proof (and thi would till be the cae without a nonuniform exponential behavior or an arbitrary function ρ). The trategy i to conider operator obtained from taking formally the derivative with repect to ξ in the two equation in (17), and then how that thee have fixed point in an appropriate pace of continuou function. The final tep i to how that thee fixed point are indeed obtained from derivating φ, and thu φ mut be of cla C 1. The main difficulty that prevent one to how that the new operator conidered together with T in (18) are contraction i that we would need further regularity aumption on f, and thu we would looe the optimal regularity of the center manifold. Intead we ue the fiber contraction principle, which require weaker aumption. Neverthele, one till need to etablih the continuity of the fiber contraction. Thi turn out to be the main technical difficulty of the proof.

10 10 Lui Barreira and Claudia Vall We conider the pace F of continuou function Φ: { (, ξ) R X : ξ E() } R L(), where L() i the family of linear tranformation from E() to F 1 () F 2 (), uch that Φ(, ξ) L() for every (, ξ) R E(), with Φ := up { Φ(, ξ) : (, ξ) R E() } 1. (19) We alo conider the ubet F 0 F compoed of the function Φ F uch that Φ(, 0) = 0 for every R. Clearly, F and F 0 are complete metric pace with the ditance induced by the norm in (19). We define a linear operator for each (φ, Φ) X F by A(φ, Φ) = (A 1 (φ, Φ), A 2 (φ, Φ)) A 1 (φ, Φ)(, ξ) ( f = Q 1 ()T (τ, ) 1 x (y φ(τ))w (τ) + f ) y (y φ(τ))φ(z φ (τ))w (τ) dτ, and A 2 (φ, Φ)(, ξ) = Q 2 ()T (τ, ) 1 ( f x (y φ(τ))w (τ) + f y (y φ(τ))φ(z φ (τ))w (τ) where (x, y) E(τ) (F 1 (τ) F 2 (τ)), with the notation y φ (t) = (t, x φ (t, ξ), φ(t, x φ (t, ξ))) and z φ (t) = (t, x φ (t, ξ)), and where the linear operator are determined by the identitie W (t) = W φ,φ,ξ (t): E() E(t) (20) ) dτ, (21) W (t) = P (t)t (t, ) ( f + P (t)t (t, τ) x (y φ(τ))w (τ) + f ) y (y φ(τ))φ(z φ (τ))w (τ) dτ (22) for t R. It follow from the continuity of the olution of a differential equation with repect to parameter, and the continuity of the function (t, φ,, ξ) x φ (t, ξ), φ, and Φ, that the function (t, φ,, ξ) W φ,φ,ξ (t) i alo continuou.

11 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 11 Lemma 5. The operator A i well-defined, and A(X F) F. Proof of the lemma. Set ( B 1 = f Q 1()T (τ, ) 1 x (y φ(τ))w (τ)+ f ) y (y φ(τ))φ(z φ (τ))w (τ) dτ, and B 2 = ( f Q 2()T (τ, ) 1 x (y φ(τ))w (τ)+ f ) y (y φ(τ))φ(z φ (τ))w (τ) dτ. It follow from (6) and (7) that B 1 2δD = 2δD ρ (τ)e a(ρ() ρ(τ))+ε ρ(τ) 3ε ρ(τ) W (τ) dτ ρ (τ)e a(ρ() ρ(τ)) 2ε ρ(τ) W (τ) dτ. Furthermore, it follow from (5) and (7) that B 2 2δD = 2δD ρ (τ)e b(ρ(τ) ρ())+ε ρ(τ) 3ε ρ(τ) W (τ) dτ ρ (τ)e b(ρ(τ) ρ()) 2ε ρ(τ) W (τ) dτ. On the other hand, by (22) and again (7), for t we have W (t) De c(ρ(t) ρ())+ε ρ() + 2δD Setting Γ(t) = e c(ρ(t) ρ()) W (t) we obtain ρ (τ)e c(ρ(t) ρ(τ))+ε ρ(τ) 3ε ρ(τ) W (τ) dτ. Γ(t) De ε ρ() + 2δD De ε ρ() + 2δD It follow from Gronwall lemma that ρ (τ)e 2ε ρ(τ) Γ(τ) dτ ρ (τ)γ(τ) dτ. Γ(t) De ε ρ() e 2δD(ρ(t) ρ()), (23) (24) (25) and thu, W (t) De ε ρ() e (c+2δd)(ρ(t) ρ()), t. (26) Proceeding in a imilar manner, we find that W (t) De ε ρ() e ( c+2δd)(ρ() ρ(t)), t. (27)

12 12 Lui Barreira and Claudia Vall Indeed, by (22) and again (7), for t we have W (t) De c(ρ() ρ(t))+ε ρ() + 2δD Setting Γ(t) = e c(ρ() ρ(t)) W (t) we obtain t Γ(t) De ε ρ() + 2δD ρ (τ)e c(ρ(τ) ρ(t))+ε ρ(τ) 3ε ρ(τ) W (τ) dτ. De ε ρ() + 2δD By Gronwall lemma we have t t ρ (τ)e 2ε ρ(τ) Γ(τ) dτ ρ (τ)γ(τ) dτ. Γ(t) De ε ρ() e 2δD(ρ() ρ(t)), and thu, inequality (27) hold. It follow from (27) and (23) that B 1 2δD 2 ρ (τ)e (a c+ε+2δd)(ρ() ρ(τ)) dτ = 2δD 2 a c + ε + 2δD < 1, (28) provided that δ i ufficiently mall. Furthermore, it follow from (26) and (24) that B 2 2δD 2 ρ (τ)e ( b+c+ε+2δd)(ρ(τ) ρ()) dτ = 2δD 2 b + c + ε + 2δD < 1, (29) provided that δ i ufficiently mall. Thi how that A(φ, Φ) i well-defined. Again for δ ufficiently mall, it follow from (28) and (29) that A(φ, Φ)(, ξ) B 1 + B 2 < 1 for every (, ξ) R E(). Thi how that A(X F) F. When ( f/ u)(t, 0) = 0 for every t R, ince x φ (t, 0) = 0 for φ X and t R, it follow from (20) and (21) that A(φ, Φ)(, 0) = 0 for every (φ, Φ) X F 0 and R. Therefore, in thi cae, A(X F 0 ) F 0. Now we conider the tranformation S : X F X F defined by with the operator T a in (18). S(φ, Φ) = (T φ, A(φ, Φ)),

13 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 13 Lemma 6. For every ufficiently mall δ > 0, the operator S i a fiber contraction. Proof of the lemma. Given R, ξ E(), φ X, and Φ, Ψ F, let W Φ = W φ,φ,ξ and W Ψ = W φ,ψ,ξ. Setting α = Φ Ψ, we obtain A 1 (φ, Φ)(, ξ) A 1 (φ, Ψ)(, ξ) D δd δd e a(ρ() ρ(τ))+ε ρ(τ) f x W Φ + f y ΦW Φ f x W Ψ f y ΨW Ψ dτ ρ (τ)e a(ρ() ρ(τ)) 2ε ρ(τ) ( W Φ W Ψ + ΦW Φ ΨW Ψ ) dτ ρ (τ)e a(ρ() ρ(τ)) 2ε ρ(τ) ( W Φ W Ψ + Φ W Φ W Ψ + α W Ψ ) dτ δd ρ (τ)e a(ρ() ρ(τ)) 2ε ρ(τ) ( 2 W Φ W Ψ + α W Ψ ) dτ, (30) and imilarly, A 2 (φ, Φ)(, ξ) A 2 (φ, Ψ)(, ξ) D e b(ρ(τ) ρ())+ε ρ(τ) f x W Φ + f y ΦW Φ f x W Ψ f y ΨW Ψ dτ δd δd ρ (τ)e b(ρ(τ) ρ()) 2ε ρ(τ) ( W Φ W Ψ + ΦW Φ ΨW Ψ ) dτ ρ (τ)e b(ρ(τ) ρ()) 2ε ρ(τ) ( 2 W Φ W Ψ + α W Ψ ) dτ, where for implicity we have omitted the argument inide the integral.

14 14 Lui Barreira and Claudia Vall In an analogou manner to that in (25) and uing (26), for t we have W Φ (t) W Ψ (t) 2δD + δd Φ Ψ ρ (τ)e c(ρ(t) ρ(τ))+ε ρ(τ) 3ε ρ(τ) W Φ (τ) W Ψ (τ) dτ ρ (τ)e c(ρ(t) ρ(τ))+ε ρ(τ) 3ε ρ(τ) W Ψ (τ) dτ 2δDe c(ρ(t) ρ()) ρ (τ)e c(ρ(τ) ρ()) 2ε ρ(τ) W Φ (τ) W Ψ (τ) dτ + δd 2 e c(ρ(t) ρ()) Φ Ψ ρ (τ)e (c+ε)(ρ(τ) ρ()) e (c+2δd)(ρ(τ) ρ()) dτ = 2δDe c(ρ(t) ρ()) ρ (τ)e c(ρ(τ) ρ()) 2ε ρ(τ) W Φ (τ) W Ψ (τ) dτ + δd 2 e c(ρ(t) ρ()) Φ Ψ Setting we thu obtain Γ(t) ρ (τ)e (ε 2δD)(ρ(τ) ρ()) dτ. Γ(t) = e c(ρ(t) ρ()) W Φ (t) W Ψ (t), δd2 Φ Ψ + 2δD ε 2δD ρ (τ)γ(τ) dτ, provided that δ i ufficiently mall. It follow from Gronwall lemma that W Φ (t) W Ψ (t) δd2 ε 2δD Φ Ψ e(c+2δd)(ρ(t) ρ()). (31) Proceeding in a imilar manner we find that for t, W Φ (t) W Ψ (t) δd2 ε 2δD Φ Ψ e( c+2δd)(ρ() ρ(t)). (32) By (26) and (31), it follow from (30) that A 1 (φ, Φ)(, ξ) A 1 (φ, Ψ)(, ξ) K 1 δ Φ Ψ K 1 δ Φ Ψ, a c + ε + 2δD ρ (τ)e (a c+ε+2δd)(ρ() ρ(τ)) dτ (33)

15 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 15 for ome contant K 1 > 0, provided that δ i ufficiently mall. Similarly, by (27) and (32), A 2 (φ, Φ)(, ξ) A 2 (φ, Ψ)(, ξ) K 2 δ Φ Ψ K 1 δ Φ Ψ, c b + ε + 2δD ρ (τ)e (c b+ε+2δd)(ρ(τ) ρ()) dτ (34) provided that δ i ufficiently mall. By (33) and (34), eventually making δ ufficiently maller, the operator S become a fiber contraction. Finally, we etablih the continuity of the fiber contraction. Lemma 7. For every ufficiently mall δ > 0, the operator S i continuou. Proof of the lemma. Setting we obtain and W φ = W φ,φ,ξ and W ψ = W ψ,φ,ξ, A 1 (φ, Φ)(, ξ) A 1 (ψ, Φ)(, ξ) D f x (y φ(τ))w φ (τ) + f y (y φ(τ))φ(z φ (τ))w φ (τ) f x (y ψ(τ))w ψ (τ) f y (y ψ(τ))φ(z ψ (τ))w ψ (τ) dτ, e a(ρ() ρ(τ))+ε ρ(τ) A 2 (φ, Φ)(, ξ) A 2 (ψ, Φ)(, ξ) e b(ρ(τ) ρ())+ε ρ(τ) D f x (y φ(τ))w φ (τ) + f y (y φ(τ))φ(z φ (τ))w φ (τ) f x (y ψ(τ))w ψ (τ) f y (y ψ(τ))φ(z ψ (τ))w ψ (τ) dτ.

16 16 Lui Barreira and Claudia Vall It follow from (7) and (27) that A 1 (φ, Φ)(, ξ) A 1 (ψ, Φ)(, ξ) D e a(ρ() ρ(τ))+ε ρ(τ) f x (y φ(τ)) f x (y ψ(τ)) W φ (τ) dτ + D e a(ρ() ρ(τ))+ε ρ(τ) f x (y ψ(τ)) W φ (τ) W ψ (τ) dτ + D e a(ρ() ρ(τ))+ε ρ(τ) f y (y φ(τ)) f y (y ψ(τ)) Φ(z φ (τ))w φ (τ) dτ + D e a(ρ() ρ(τ))+ε ρ(τ) f y (y ψ(τ)) Φ(z φ (τ)) Φ(z ψ (τ)) W φ (τ) dτ + D e a(ρ() ρ(τ))+ε ρ(τ) f y (y ψ(τ)) Φ(z ψ (τ)) W φ (τ) W ψ (τ) dτ, and thu, A 1 (φ, Φ)(, ξ) A 1 (ψ, Φ)(, ξ) D 2 e 2ε ρ() e ( c+2δd ε+a)(ρ() ρ(τ)) f x (y φ(τ)) f + δd ρ (τ)e a(ρ(τ) ρ()) 2ε ρ(τ) W φ (τ) W ψ (τ) dτ + D 2 e 2ε ρ() + δd 2 + δd e ( c+2δd ε+a)(ρ() ρ(τ)) f y (y φ(τ)) f x (y ψ(τ)) dτ y (y ψ(τ)) dτ ρ (τ)e ( c+2δd+ε+a)(ρ() ρ(τ)) ε ρ(τ) Φ(z φ (τ)) Φ(z ψ (τ)) dτ ρ (τ)e a(ρ() ρ(τ)) 2ε ρ(τ) W φ (τ) W ψ (τ) dτ

17 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 17 2D 2 e 2ε ρ() e ( c+2δd ε+a)(ρ() ρ(τ)) f u (y φ(τ)) f u (y ψ(τ)) dτ + 2δD ρ (τ)e a(ρ() ρ(τ)) 2ε ρ(τ) W φ (τ) W ψ (τ) dτ + δd 2 ρ (τ)e ( c+2δd+ε+a)(ρ() ρ(τ)) ε ρ(τ) Φ(z φ (τ)) Φ(z ψ (τ)) dτ. (35) Moreover, given γ > 0 there exit σ > 0 (independent of and ξ) uch that, etting η = ρ 1 (ρ() ρ(σ)), η 2D 2 e 2ε ρ() e ( c+2δd ε+a)(ρ() ρ(τ)) f u (y φ(τ)) f u (y ψ(τ)) dτ η 4δD 2 ρ (τ)e ( c+2δd+ε+a)(ρ() ρ(τ)) dτ and = 4δD2 e ( c+2δd+ε+a)ρ(σ) c + 2δD + ε + a δd 2 η < γ, η 2δD ρ (τ)e a(ρ() ρ(τ)) 2ε ρ(τ) W φ (τ) W ψ (τ) dτ η 4δD 2 ρ (τ)e ( c+2δd+ε+a)(ρ() ρ(τ)) dτ < γ, 2δD 2 η ρ (τ)e ( c+2δd+ε+a)(ρ() ρ(τ)) ε ρ(τ) Φ(z φ (τ)) Φ(z ψ (τ)) dτ ρ (τ)e ( c+2δd+ε+a)(ρ() ρ(τ)) dτ < γ. (36) (37) (38) Now we conider the integral from η to. For thi, etting p = ρ() ρ(τ) we conider the function B(p, φ)(, ξ) = 2D2 e 2ε ρ() e ( c+2δd ε+a)p ρ (ρ 1 (ρ() p)) C(p, φ)(, ξ) = 2δDe ap 2ε ρ() p W φ (ρ 1 (ρ() p)), f u (y φ(ρ 1 (ρ() p))), D(p, φ)(, ξ) = δd 2 e ( c+2δd+ε+a)p ε ρ() p Φ(z φ (ρ 1 (ρ() p)))

18 18 Lui Barreira and Claudia Vall for each p [0, ρ(σ)] and φ X. By (35), it i ufficient to how that the map ρ(σ) [ ] φ B(p, φ) + C(p, φ) + D(p, φ) dp i continuou. Since the function Φ, 0 (t, φ,, ξ) x φ (t, ξ), and (t, φ,, ξ) W φ,φ,ξ (t) are continuou, the function (p, φ,, ξ) B(p, φ)(, ξ), C(p, φ)(, ξ), D(p, φ)(, ξ) (39) are alo continuou. Furthermore, by (7) and (27), for each p [0, ρ(σ)] and φ X we have B(p, φ) 2δD 2 e ( c+2δd+ε+a)p ε ρ() p 2δD 2 e ε ρ() p, C(p, φ) 2δD 2 e ( c+2δd+ε+a)p ε ρ() p 2δD 2 e ε ρ() p, D(p, φ) δd 2 e ( c+2δd+ε+a)p ε ρ() p δd 2 e ε ρ() p, with the norm in (19). In particular, B(p, φ), C(p, φ), and D(p, φ) are in F provided that δ i ufficiently mall. We firt note that there exit R > 0 uch that B(p, φ)(, ξ) B(p, ψ)(, ξ) 4δD 2 e ε ρ() p < γ, C(p, φ)(, ξ) C(p, ψ)(, ξ) 4δD 2 e ε ρ() p < γ, D(p, φ)(, ξ) D(p, ψ)(, ξ) 4δD 2 e ε ρ() p < γ for every > R, p [0, ρ(σ)], and ξ E(). It remain to conider the cae when R. Given R and (φ, ψ) X E(), due to the continuity in (39) there exit δ > 0 uch that B(p, φ)(, ξ) B(q, ψ)(, ξ) < γ whenever d(φ, ψ) < δ and (p,, ξ) (q,, ξ) < δ. Since u f(t, u) vanihe for u c, it i ufficient to etablih the deired continuity for ξ inide a certain ball in E(), poibly depending (continuouly) on p and. Thi how that it i ufficient to conider ξ in ome compact et K. We can cover [0, ρ(σ)] [ R, R] K with a finite number of open ball B i, i = 1,..., r centered at point in thi et, uch that B(p, φ)(, ξ) B( p, ψ)(, ξ) < γ whenever d(φ, ψ) < δ i and (p,, ξ), ( p,, ξ) B i, for i = 1,..., r and ome number δ i > 0. Therefore, B(p, φ)(, ξ) B(p, ψ)(, ξ) < γ

19 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 19 whenever d(φ, ψ) < δ = min{δ 1,..., δ r }, for every p [0, ρ(σ)], R, and ξ K. Thi how that up R ξ K up B(p, φ)(, ξ) B(p, ψ)(, ξ) γ whenever d(φ, ψ) < δ and p [0, ρ(σ)]. Similar argument apply to C(p, φ) and D(p, φ). Together with (36), (37), and (38) thi implie that φ A 1 (φ, Φ) i continuou. Similarly, it follow from (7) and (26) that A 2 (φ, Φ)(, ξ) A 2 (ψ, Φ)(, ξ) D e b(ρ(τ) ρ())+ε ρ(τ) f x (y φ(τ)) f x (y ψ(τ)) W φ (τ) dτ + D e b(ρ(τ) ρ())+ε ρ(τ) f x (y ψ(τ)) W φ (τ) W ψ (τ) dτ + D e b(ρ(τ) ρ())+ε ρ(τ) f y (y φ(τ)) f y (y ψ(τ)) Φ(z φ (τ))w φ (τ) dτ + D e b(ρ(τ) ρ())+ε ρ(τ) f y (y ψ(τ)) Φ(z φ (τ)) Φ(z ψ (τ)) W φ (τ) dτ + D e b(ρ(τ) ρ())+ε ρ(τ) f y (y ψ(τ)) Φ(z ψ (τ)) W φ (τ) W ψ (τ) dτ. Proceeding in a imilar manner to that for A 1 we find that φ A 2 (φ, Φ) i continuou, and thu φ A(φ, Φ) i alo continuou. Thi how that the fiber contraction S i continuou (we already know that the operator T in (18) i a contraction, and thu it i alo continuou). To etablih the C 1 regularity, we proceed with an auxiliary tatement. Lemma 8. If φ i of cla C 1 in ξ, then T φ i of cla C 1 in ξ, and (T φ)/ ξ = A(φ, φ/ ξ). (40) Proof of the lemma. Since φ i of cla C 1 in ξ, the function y defined by y(t, ξ) = x φ (t, ξ) i alo of cla C 1 (the right-hand ide of (16) i of cla C 1, and thu the olution are C 1 in the initial condition). Furthermore, for Φ = φ/ ξ the olution of equation (22) i given by W (t) = y/ ξ.

20 20 Lui Barreira and Claudia Vall Therefore, repeating argument in the proof of Lemma 5 we can apply Leibnitz rule to obtain A(φ, φ/ ξ)(, ξ) ( = ξ ξ [ Q 1 ()T (τ, ) 1 f(τ, x φ (τ, ξ), φ(τ, x φ (τ, ξ))) [ Q 2 ()T (τ, ) 1 f(τ, x φ (τ, ξ), φ(τ, x φ (τ, ξ))) = ( (T φ)/ ξ ) (, ξ), for every R and ξ E(). ] dτ, ] ) dτ We conider the pair (φ 1, Φ 1 ) = (0, 0) X F. Clearly, Φ 1 = φ 1 / ξ. We define recurively a equence (φ n, Φ n ) X F by (φ n+1, Φ n+1 ) = S(φ n, Φ n ) = (T φ n, A(φ n, Φ n )). (41) Auming that φ n i of cla C 1 in ξ with Φ n = φ n / ξ, it follow from Lemma 8 that φ n+1 = T φ n i of cla C 1 in ξ, and by (40) we have φ n+1 / ξ = (T φ n )/ ξ = A(φ n, Φ n ) = Φ n+1. (42) Now let φ 0 be the unique fixed point of T, and let Φ 0 be the unique fixed point of Ψ A(φ 0, Ψ). By Propoition 1 the equence φ n and Φ n converge uniformly repectively to φ 0 and Φ 0 on bounded ubet. It follow from (42) that φ 0 i of cla C 1 in ξ, and that φ 0 / ξ = Φ 0 (43) (we recall that if a equence f n of C 1 function converge uniformly, and the equence f n of it derivative alo converge uniformly, then the limit of f n i of cla C 1, and it derivative i the limit of f n). Now we aume that ( f/ u)(t, 0) = 0 for every t R. Since the pair (φ 1, Φ 1 ) = (0, 0) i in X F 0, and S(X F 0 ) X F 0, the equence (φ n, Φ n ) defined in (41) i alo in X F 0. Therefore, Φ 0 (, 0) = 0 for every R, and it follow from (43) that in thi cae ( φ 0 / ξ)(, 0) = 0 for every R. Finally, we how that the et V φ in (11) i a C 1 manifold. We already know that the function ξ φ(, ξ) i of cla C 1 for each fixed R. We define a map F = F : R E() R X by F (t, ξ) = Ψ t (, ξ, φ(, ξ)).

21 Center Manifold Optimal Regularity for Nonuniformly Hyperbolic Dynamic 21 Since A and f are of cla C 1, the map R R X (t,, v) Ψ t (, v) i alo of cla C 1, and the ame happen with F. We can eaily verify that F i injective, and thu it i a parametrization of cla C 1 of V φ. When ( f/ u)(t, 0) = 0 for every t R, the firt identity in (13) yield the econd one. Reference 1. L. Barreira and Ya. Pein, Nonuniform Hyperbolicity, Encyclopedia of Math. and It Appl. 115, Cambridge Univ. Pre, L. Barreira and C. Vall, Smooth center manifold for nonuniformly partially hyperbolic trajectorie, J. Differential Equation 237 (2007), L. Barreira and C. Vall, Stability of Nonautonomou Differential Equation, Lect. Note in Math. 1926, Springer, L. Barreira and C. Vall, Center manifold for nonuniform trichotomie and arbitrary growth rate, Comm. Pure Appl. Anal., to appear. 5. L. Barreira and C. Vall, Regularity of center manifold under nonuniform hyperbolicity, preprint. 6. J. Carr, Application of Centre Manifold Theory, Applied Mathematical Science 35, Springer, C. Chicone, Ordinary Differential Equation with Application, Text in Applied Mathematic 34, Springer, C. Chicone and Yu. Latuhkin, Center manifold for infinite dimenional nonautonomou differential equation, J. Differential Equation 141 (1997), S.-N. Chow, W. Liu and Y. Yi, Center manifold for invariant et, J. Differential Equation 168 (2000), S.-N. Chow, W. Liu and Y. Yi, Center manifold for mooth invariant manifold, Tran. Amer. Math. Soc. 352 (2000), S.-N. Chow and K. Lu, C k centre untable manifold Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), C. Contantine and T. Savit, A multivariate Faà di Bruno formula with application, Tran. Amer. Math. Soc. 348 (1996), M. Elbialy, On equence of C k,δ b map which converge in the uniform C 0 -norm, Proc. Amer. Math. Soc. 128 (2000), D. Henry, Geometric Theory of Semilinear Parabolic Equation, Lect. Note in Math. 840, Springer, A. Kelley, The table, center-table, center, center-untable, untable manifold, J. Differential Equation 3 (1967), A. Mielke, A reduction principle for nonautonomou ytem in infinitedimenional pace, J. Differential Equation 65 (1986), V. Oeledet, A multiplicative ergodic theorem. Liapunov characteritic number for dynamical ytem, Tran. Mocow Math. Soc. 19 (1968), Ya. Pein, Familie of invariant manifold correponding to nonzero characteritic exponent, Math. USSR-Izv. 10 (1976), V. Pli, A reduction principle in the theory of tability of motion, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), A. Vanderbauwhede, Centre manifold, normal form and elementary bifurcation, in Dynamic Reported 2, Wiley, 1989, pp

22 22 Lui Barreira and Claudia Vall 21. A. Vanderbauwhede and G. Ioo, Center manifold theory in infinite dimenion, in Dynamic Reported (N.S.) 1, Springer, 1992, pp A. Vanderbauwhede and S. van Gil, Center manifold and contraction on a cale of Banach pace, J. Funct. Anal. 72 (1987),