We present an example of a smooth invertible contraction in aninfinite-dimensional Hilbert space that is not locally ${\mathcalC}^{1}$-linearizable near its fixed point.Nous présentons un exemple de contraction inversible et régulière dans un espace de Hilbert de dimension infinie qui n'est pas localement C1-linéarisable autour de son point fixe

Munhoz Rodrigues, Hildebrando

Solà-Morales Rubió, Joan de

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An Invertible Contraction that is not C1-linearizableUne Contraction Inversible qui n’est pas C1-line´arisableHildebrando M. Rodrigues a,1 J. Sola`-Morales b,2aDepartamento de Matema´tica, Instituto de Cieˆncias Matema´ticas e de Computac¸a˜o, Universidade de Sa˜o Paulo, CaixaPostal 668, 13560-970, Sa˜o Carlos, SP, BrazilbDepartament de Matema`tica Aplicada 1, Universitat Polite`cnica de Catalunya, Av. Diagonal 647, 08028 Barcelona,SpainAbstractWe present an example of a smooth invertible contraction in an infinite-dimensional Hilbert space that is notlocally C1-linearizable near its fixed point. To cite this article: H.M. Rodrigues, J.Sola`-Morales, C. R. Acad. Sci.Paris, Ser. I xxx (xxxx).Re´sume´Nous pre´sentons un exemple de contraction inversible et re´gulie`re dans un espace de Hilbert de dimension infiniequi n’est pas localement C1-line´arisable autour de son point fixe. Pour citer cet article : H.M. Rodrigues, J.Sola`-Morales, C. R. Acad. Sci. Paris, Ser. I xxx (xxxx).1. Introduction and Main Result.An invertible contraction both in finite or infinite dimensions can always be linearized in the class C0,by the well known Hartman-Grobman theorem, as it was proved in Pugh [7]. But it seems that it wasnot known until now if this was also true for the linearization in the class C1 (see Abbaci [2]) in the caseof infinite dimensional Banach spaces. All the existing results on this case, that to our knowledge arethose of Mora, Sola`-Morales [6], Tan [11], and the three independent recent works ElBialy [3], AbbaciEmail addresses: hmr@icmc.usp.br (Hildebrando M. Rodrigues), jc.sola-morales@upc.edu (J. Sola`-Morales).1 Partially supported by Fapesp Processo : 04/08597-7, CNPq Processo : 301994/85-4, CAPES/MECD 023/01 and Pro-grama Pronex-Projeto Tematico, CNPq-FAPESP no. 2003/10042-0, Brasil.2 Partially supported by MCyT-MEC, Spain (BMF2002-04613-C03-01).Preprint submitted to Elsevier Science 13th April 2005[1], [2] and Rodrigues, Sola`-Morales [9] require extra hypotheses for an invertible contraction in infinitedimensions to be C1-linearizable. (We take this opportunity to apologize for having written our work [9]without knowledge of the existence of [3] and [1].) For example, in the paper [9] we proved the followingtheorem:A Linearization Theorem For Contractions. Let X be a Banach space with the property thatthere exists a function ρ such thatρ ∈ C1,1(X,R), with ρ(z) = 1, when |z| ≤ 1/2 and ρ(z) = 0, when |z| ≥ 1. (1)Suppose that L, L−1 ∈ L(X). We assume that there exist real numbers ν−i , ν+i , i = 1, · · · , n such that0 < ν−n < ν+n < ν−n−1 < ν+n−1 < · · · < ν−1 < ν+1 < 1,ν+1 ν+i < ν−i , i = 1, · · · , n (nonresonance condition) (2)and |σ(L)| ⊂ (ν−n , ν+n ) ∪ (ν−n−1, ν+n−1) ∪ · · · ∪ (ν−1 , ν+1 ).Let F = F (z) be a C1,1-function in a neighborhood of the origin with values in X, such that F =0, DF = 0, at z = 0.Then, for the map T : z 7→ z′, z′ = Lz + F (z), there exists a C1-map R : z 7→ u, u = z + ψ(z),satisfying ψ = 0, Dψ = 0, at z = 0, such that RTR−1 : u 7→ u′ has the form u′ = Lu in a sufficientlysmall neighborhood of the origin.In the present Note we exhibit an example of a smooth invertible contraction that is not C1-linearizable.Our Banach space X will be the usual Hilbert space `2 of the square sumable sequences, that obviouslysatisfies the condition (1). The linear operator L is an invertible contraction and the nonlinearity F (z)will be a polynomial of degree 2. Also the set |σ(L)| will consist of a single interval, and will not satisfythe nonresonance condition (2).To our knowledge, this example appears to be the first one of this kind. It shows that the infinite-dimensional case is not like the finite-dimensional one, where all smooth invertible contractions can belinearized with a linearization of class C1 (see Hartman [5], or Chicone, Swanson [4]). Our example closesthis question. As a consequence, to linearize a smooth invertible contraction in the class C1 in the infinitedimensional case, one can not avoid an extra hypothesis, perhaps like the nonresonance condition (2).Our interest in the linearization problems started years ago, in the works Mora, Sola`-Morales [6] andRodrigues, Ruas [8]. Recently, we have been working in C1-linearization in infinite dimensions, in theworks Rodrigues, Sola`-Morales [9], for the case of invertible contractions, and Rodrigues, Sola`-Morales[10] where a case of a saddle point is studied. In both cases, applications to abstract wave equations havebeen presented. The present Note is a continuation of these previous works.The main idea of our example appears in the following proposition. As the reader can appreciate, ifone takes δ > |a2 − a| and one makes the dimension n to grow unboundedly then the invariant manifoldwill grow without bound.Proposition: Let 0 < a < 1 and ε, δ ∈ R be positive numbers. Consider the map (x, ξ1, . . . , ξn) 7→(x′, ξ′1, . . . , ξ′n) in Rn+1 defined by:x′ = ax, ξ′k = aξk + δξk+1, for k = 1, 2 . . . n− 1, and, ξ′n = aξn + εx2. (3)If ξi = φi(x), i = 1, . . . , n, defines a local invariant curve for the above map, differentiable at x =0, such that φi(0) = 0, φ′i(0) = 0, then φi(x) = δn−iεx2/(a2 − a)n−i+1, and in particular |φ1(x)| ≥δn−1εx2/|a2 − a|n.Let us introduce some notation. Let us write yn := (yn,1, . . . , yn,n) for a generic vector of Rn, anddefine the linear map Jn : Rn → Rn by Jnyn := (yn,2, . . . , yn,n, 0). Let In be the identity in Rn. We willconsider the linear map Ln := aIn + δJn for some given scalars a, δ.2Let us also write z := (x,y2,y3, . . . ,yn, . . .) for a generic vector of `2, and define L : `2 → `2 by Lz :=(ax, L2y2, L3y3, . . . , Lnyn, . . .). Define also the quadratic maps fn : R → Rn by fn(x) = (0, 0, . . . x2) andfinally F : `2 → `2 by F (z) := (0, ε2f2(x), ε3f3(x), . . . , εnfn(x), . . .) for a given sequence (εn).The following is our main result:Theorem: Let ε > 0 and εn := ε/n. Under the hypothesis,0 < a < 1, a− a2 < δ < min{1− a, a} (4)the operator L is a contraction on `2, satisfies |σ(L)| = [a − δ, a + δ], and the polynomial map of degree2 defined in `2 byz′ = Lz + F (z),is not C1-linearizable in any neighborhood of z = 0.Remark 1: Observe that (4) implies that (a + δ)2 > a − δ, so the nonresonance condition (2) is notsatisfied.Remark 2: To prove the Theorem we will call T := L+ F and we will suppose that a local invertiblemap R exists such that R and R−1 are of class C1 with RTR−1 = L, and then we will arrive to acontradiction. But following carefully the proof one can see that to arrive to a contradiction we do notneed even to require R and R−1 to be of class C1, but merely R and R−1 to be differentiable at z = 0.2. Proof of the Theorem.Lemma: If 0 < a < 1 and r ∈ R, then the functional equation,φ(ax) = aφ(x) + rx2 (5)has a unique local solution φ differentiable at x = 0, with φ(0) = 0 and φ′(0) = 0. This solution is givenby φ(x) = rx2/(a2 − a).Proof: Since rx2/(a2 − a) is a particular solution, the other solutions would be of the form φ(x) =rx2/(a2 − a) + φ1(x), where φ1 satisfies the homogeneous equation:φ1(ax) = aφ1(x).Let x0 6= 0. Then φ1(anx0) = anφ1(x0). Since φ1(0) = 0 and anx0 → 0 as n→∞ we haveφ′1(0) = limn→∞φ1(anx0)− φ1(0)anx0=φ1(x0)x0.So φ1(x0) = 0, since φ′1(0) = 0.Proof of the Proposition:We consider the functional equations satisfied by the φi. Starting with i = n and using the previousLemma one obtains ξn = φn(x) = εx2/(a2 − a). Substituting this expression in the n − 1 equationξ′n = aξn−1 + δξn, using again the Lemma one obtains:ξn−1 = φn−1(x) =εδ(a2 − a)2x2.Proceeding recursively one finally obtains: ξ1 = φ1(x) = εδn−1x2/(a2 − a)n.Proof of the Theorem:Let us call T := L+ F , and suppose that a local linearization map R exists such that RTR−1 = L. Ifboth R and R−1 are differentiable at zero then from RTR−1 = L one obtains that DR(0)L = LDR(0)and so DR(0)−1RT (DR(0)−1R)−1 = L. So we can suppose that DR(0) = I.3Now, the linear subspace {(x, 0)} ⊂ `2 is invariant by L, so R−1{(x, 0)} is invariant by T . This invariantcurve can be expressed as {x,Φ(x)} ⊂ `2 in a neighborhood of zero, with Φ(0) = 0, DΦ(0) = 0.Let y = (y2,y3, · · · ,yn, · · ·), with yn being as before a vector with n components, and let us writealso Φ(x) = (φ2(x), φ3(x), · · ·). It is clear that yn = φn(x) will be an invariant manifold for the n + 1-dimensional system (3), with ε = εn. Then, because of the Proposition, we have‖Φ(x)‖ ≥εnδn−1|a2 − a|nx2and we obtain a contradiction by letting n→∞ if δ > |a2 − a|, as it was taken in (4).Observe also that ‖Ln‖ = ‖aIn + δJn‖ ≤ a‖In‖+ δ‖Jn‖ = a+ δ.Since this bound is independent of n we get that ‖L‖ ≤ a+ δ, and because of (4), L is a contraction.Next, one can show that the spectrum of L is the whole disk |z − a| ≤ δ of the complex plane. We donot give all the details, but we merely say that this can be easily deduced from the following estimates:Let c ∈ C and In, Jn defined as above. If |c| > 1, then ‖(cIn + Jn)−1‖ ≤ 1/(|c| − 1), and if 0 < |c| < 1,then ‖(cIn + Jn)−1‖ ≥ 1/|c|n, for all n ≥ 2. These estimates follow easily from the explicit formula(cIn + Jn)−1 =1cIn −1c2Jn +1c3J2n + · · ·+(−1)n−1cnJn−1n .Remark 3:In order to obtain the slightly better result described in Remark 2, the previous proof requiresa small modification: one has to prove that the set R−1{(x, 0)} is can be expressed as {x,Φ(x)}. To dothat one can write it as {(x+ φ1(x)),Φ(x)} and then prove that φ1 ≡ 0, by using the Lemma with r = 0.References[1] Abbaci, B. Variete´s Invariantes et Applications, The`se, Universite´ Paris 7, 2001.[2] Abbaci, B. On a Theorem of Philip Hartman, C.R. Acad. Sci. Paris, Ser. I, 339, (2004) 781-786.[3] ElBialy, M. S. Local Contractions of Banach Spaces and Spectral Gap Conditions, Journal of Functional Analysis, 182,(2001) 108-150.[4] Chicone, C. and Swanson, R. Linearization via the Lie Derivative, Electron. J. Diff. Eqns. Monograph, 02 (2000).[5] Hartman, P. On local homeomorphisms of euclidean spaces, Boletin de la Sociedad Matematica Mexicana, 5, 2, (1960)220-241.[6] Mora, X., Sola`-Morales, J.. Existence and Non-existence of Finite-dimensional Globally Attracting invariant Manifoldsin Semilinear Damped Wave Equations, in Dynamics of infinite-dimensional systems (Lisbon, 1986), 187-210, NATOAdv. Sci. Inst. Ser. F: Comput. Systems Sci., 37, Springer, Berlin-New York (1987) 187-210.[7] Pugh, C.C. On a Theorem of P. Hartman, Amer. J. Math. 91 (1969) 363-367.[8] Rodrigues, H. M., Ruas Filho, J.G.. The Hartman-Grobman Theorem for Reversible Systems on Banach spaces, J.Nonlinear Sci. 7 (1997) 271-280.[9] Rodrigues, H. M., Sola`-Morales, J. Linearization of Class C1 for Contractions on Banach Spaces, J. DifferentialEquations 201 (2004) 351-382.[10] Rodrigues, H. M., Sola`-Morales, J. Smooth Linearization for a Saddle on Banach Spaces, J. Dynamics and DifferentialEquations, Vol.16, No.3 (2004) 767-793.[11] Tan, B. Invariant Manifolds, Invariant Foliations and Linearization Theorems in Banach Space, PhD. Thesis, GeorgiaInstitute of Technology, (1998).4

Une contraction inversible qui n'est pas C1-linéarisable

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Une contraction inversible qui n'est pas C1-linéarisable

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