Electron. J. Diff. Eqns., Monograph 02, 2000 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp) Linearization via the Lie Derivative ∗ Carmen Chicone & Richard Swanson Abstract The standard proof of the Grobman–Hartman linearization theorem for a flow at a hyperbolic rest point proceeds by first establishing the analogous result for hyperbolic fixed points of local diffeomorphisms. In this exposition we present a simple direct proof that avoids the discrete case altogether. We give new proofs for Hartman’s smoothness results: A C2 flow is C1 linearizable at a hyperbolic sink, and a C2 flow in the plane is C1 linearizable at a hyperbolic rest point. Also, we formulate and prove some new results on smooth linearization for special classes of quasi-linear vector fields where either the nonlinear part is restricted or additional conditions on the spectrum of the linear part (not related to resonance conditions) are imposed. Contents 1 Introduction 2 2 Continuous Conjugacy 4 3 Smooth Conjugacy 7 3.1 Hyperbolic Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.1 Smooth Linearization on the Line . . . . . . . . . . . . . 32 3.2 Hyperbolic Saddles . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Linearization of Special Vector Fields 45 4.1 Special Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Saddles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Infinitesimal Conjugacy and Fiber Contractions . . . . . . . . . . 50 4.4 Sources and Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ∗Mathematics Subject Classifications: 34-02, 34C20, 37D05, 37G10. Key words: Smooth linearization, Lie derivative, Hartman, Grobman, hyperbolic rest point, fiber contraction, Dorroh smoothing. c©2000 Southwest Texas State University. Submitted November 14, 2000. Published December 4, 2000. 1 2 Linearization via the Lie Derivative 1 Introduction This paper is divided into three parts. In the first part, a new proof is presented for the Grobman–Hartman linearization theorem: A C1 flow is C0 linearizable at a hyperbolic rest point. The second part is a discussion of Hartman’s results on smooth linearization where smoothness of the linearizing transformation is proved in those cases where resonance conditions are not required. For example, we will use the theory of ordinary differential equations to prove two main theorems: A C2 vector field is C1 linearizable at a hyperbolic sink; and, a C2 vector field in the plane is C1 linearizable at a hyperbolic rest point. In the third part, we will study a special class of vector fields where the smoothness of the linearizing transformation can be improved. The proof of the existence of a smooth linearizing transformation at a hyper- bolic sink is delicate. It uses a version of the stable manifold theorem, consider- ation of the gaps in the spectrum of the linearized vector field at the rest point, carefully constructed Gronwall type estimates, and an induction argument. The main lemma is a result about partial linearization by near-identity transforma- tions that are continuously differentiable with Hölder derivatives. The method of the proof requires the Hölder exponent of these derivatives to be less than a certain number, called the Hölder spectral exponent, that is defined for linear maps as follows. Suppose that {−b1,−b2, · · ·− bN} is the set of real parts of the eigenvalues of the linear transformation A : Rn → Rn and −bN < −bN−1 < · · · < −b1 < 0. (1.1) The Hölder spectral exponent of A is the number b1(bj+1 − bj) b1(bj+1 − bj) + bj+1bj where bj+1 − bj bj+1bj = min i∈{1,2,··· ,N−1} bi+1 − bi bi+1bi in case N > 1; it is the number one in case N = 1. The Hölder spectral exponent of a linear transformation B whose eigenvalues all have positive real parts is the Hölder spectral exponent of −B. Although a C2 flow in the plane is always C1 linearizable at a hyperbolic rest point, a C2 flow in R3 may not be C1 linearizable at a hyperbolic saddle point. For example, the flow of the system ẋ = 2x, ẏ = y + xz, ż = −z (1.2) is not C1 linearizable at the origin (see Hartman’s example (3.1)). We will prove that a flow in Rn can be smoothly linearized at a hyperbolic saddle if the spectrum of the corresponding linearized system at the saddle point satisfies the following condition introduced by Hartman in [H60M]. Note first that the Carmen Chicone & Richard Swanson 3 real parts of the eigenvalues of the system matrix of the linearized system at a hyperbolic saddle lie in the union of two intervals, say [−aL,−aR] and [bL, bR] where aL, aR, bL, and bR are all positive real numbers. Thus, the system matrix can be written as a direct sum A⊕B where the real parts of the eigenvalues of A are in [−aL,−aR] and the real parts of the eigenvalues of B are in [bL, bR]. Let µ denote the Hölder spectral exponent of A and ν the Hölder spectral exponent of B. If Hartman’s spectral condition aL − aR < µbL, bR − bL < νaR is satisfied, then the C2 nonlinear system is C1 linearizable at the hyperbolic saddle point. It follows that, unlike system (1.2), the flow of ẋ = 2x, ẏ = y + xz, ż = −4z is C1 linearizable at the origin. In the case of hyperbolic saddles where one of the Hölder spectral exponents is small, Hartman’s spectral condition is satisfied only if the corresponding real parts of the eigenvalues of the linear part of the field are contained in an ac- cordingly small interval. Although the situation cannot be improved for general vector fields, stronger results (in the spirit of Hartman) are possible for a re- stricted class of vector fields. There are at least two ways to proceed: additional conditions can be imposed on the spectrum of the linearization, or restrictions can be imposed on the nonlinear part of the vector field. We will show that a C3 vector field in “triangular form” with a hyperbolic saddle point at the origin can be C1 linearized if Hartman’s spectral condition is replaced by the inequalities aL − aR < bL and bR − bL < aR (see Theorem 4.6). Also, we will prove the following result: Suppose that X = A+F is a quasi-linear C3 vector field with a hyperbolic saddle at the origin, the set of negative real parts of eigenvalues of A is given by {−λ1, . . . ,−λp}, the set of positive real parts is given by {σ1, . . . , σq}, and −λ1 < −λ2 < · · · < −λp < 0 < σq < σq−1 < · · · < σ1. If λi−1/λi > 3, for i ∈ {2, 3, . . . , p}, and σi−1/σi > 3, for i ∈ {2, 3, . . . , q}, and if λ1 − λp < σq and σ1 − σq < λp, then X is C1 linearizable (see Theorem 4.7). The important dynamical behavior of a nonlinear system associated with a hyperbolic sink is local: there is an open basin of attraction and every trajec- tory that enters this set is asymptotically attracted to the sink. This behavior is adequately explained by using a linearizing homeomorphism, that is, by using the Grobman–Hartman theorem. On the other hand, the interesting dynamical behavior associated with saddles is global; for example, limit cycles are pro- duced by homoclinic loop bifurcations and chaotic invariant sets are found near transversal intersections of homoclinic manifolds. Smooth linearizations at hy- perbolic saddle points are used to analyze these global phenomena. It turns out that results on the smooth linearization at hyperbolic sinks are key lemmas re- quired to prove the existence of smooth linearization for hyperbolic saddles. In fact, this is the main reason to study smooth linearization at hyperbolic sinks. 4 Linearization via the Lie Derivative We treat only the case of rest points here, but we expect that our method can be applied to the problem of linearization near general invariant manifolds of differential equations. Hartman’s article [H60M] is the main reference for our results on smooth- ness of linearizations. Other primary sources are the papers [G59], [H60], [H63], and [St57]. For historical remarks, additional references, and later work see [CL88], [CLL91], [KP90], [Se85], [St89], and [T99]. 2 Continuous Conjugacy A C1 vector field X on Rn such that X(0) = 0 is called locally topologically conjugate to its linearization A := DX(0) at the origin if there is a homeomor- phism h : U → V of neighborhoods of the origin such that the flows of X and A are locally conjugated by h; that is, h(etAx) = Xt(h(x)) (2.1) whenever x ∈ U , t ∈ Rn, and both sides of the conjugacy equation are defined. A matrix is infinitesimally hyperbolic if every one of its eigenvalues has a nonzero real part. Theorem 2.1 (Grobman–Hartman). Let X be a C1 vector field on Rn such that X(0) = 0. If the linearization A of X at the origin is infinitesimally hyperbolic, then X is locally topologically conjugate to A at the origin. Proof. For each r > 0 there is a smooth bump function ρ : Rn → [0, 1] with the following properties: ρ(x) ≡ 1 for |x| < r/2, ρ(x) ≡ 0 for |x| > r, and |dρ(x)| < 4/r for x ∈ Rn. The vector field Y = A + ξ where ξ(x) := ρ(x)(X(x) − Ax) is equal to X on the open ball of radius r/2 at the origin. Thus, it suffices to prove that Y is locally conjugate to A at the origin. Suppose that in equation (2.1) h = id+ η and η : Rn → Rn is differentiable in the direction A. Rewrite equation (2.1) in the form e−tAh(etAx) = e−tAXt(h(x)) (2.2) and differentiate both sides with respect to t at t = 0 to obtain the infinitesimal conjugacy equation LAη = ξ ◦ (id +η) (2.3) where LAη := d dt (e−tAη(etA)) ∣∣ t=0 (2.4) is the Lie derivative of η along A. (We note that if h is a conjugacy, then the right-hand-side of equation (2.2) is differentiable; and therefore, the Lie derivative of h in the direction A is defined.) Carmen Chicone & Richard Swanson 5 We will show that if r > 0 is sufficiently small, then the infinitesimal conju- gacy equation has a bounded continuous solution η : Rn → Rn (differentiable along A) such that h := id +η is a homeomorphism of Rn whose restriction to the ball of radius r/2 at the origin is a local conjugacy as in equation (2.1). Since A is infinitesimally hyperbolic, A = A+ ⊕ A− having spectra, respec- tively, to the left and to the right of the imaginary axis. Put E− = Range(A−) and E+ = Range(A+). There are positive constants C and λ such that |etAv+| ≤ Ce−λt|v+|, |e−tAv−| ≤ Ce−λt|v−| (2.5) for t ≥ 0. The Banach space B of bounded (in the supremum norm) continuous vector fields on Rn splits into the complementary subspaces B+ and B− of vector fields with ranges, respectively, in E+ or E−. In particular, a vector field η ∈ B has a unique representation η = η+ + η− where η+ ∈ B+ and η− ∈ B−. The function G on B defined by Gη(x) = ∫ ∞ 0 etAη+(e−tAx) dt− ∫ ∞ 0 e−tAη−(etAx) dt (2.6) is a bounded linear operator G : B → B. The boundedness of G follows from the hyperbolic estimates (2.5). The continuity of the function x 7→ Gη(x) is an immediate consequence of the following lemma from advanced calculus— essentially the Weierstrass M -test—and the hyperbolic estimates. Lemma 2.2. Suppose that f : [0,∞) × Rn → Rm, given by (t, x) 7→ f(t, x), is continuous (respectively, the partial derivative fx is continuous). If for each y ∈ Rn there is an open set S ⊂ Rn with compact closure S̄ and a function M : [0,∞) → R such that y ∈ S, the integral ∫∞ 0 M(t) dt converges, and |f(t, x)| ≤ M(t) (respectively, |fx(t, x)| ≤ M(t) ) whenever t ∈ [0,∞) and x is in S̄, then F : Rn → Rm given by F (x) = ∫∞ 0 f(t, x) dt is continuous (respectively, F is continuously differentiable and DF (x) = ∫∞ 0 fx(t, x) dt ). Using the definition of LA in display (2.4) and the fundamental theorem of calculus, we have the identity LAG = idB. As a consequence, if η = G(ξ ◦ (id +η)) := F (η), (2.7) then η is a solution of the infinitesimal conjugacy equation (2.3). Clearly, F : B → B and for η1 and η2 in B we have that ‖F (η1)− F (η2)‖ ≤ ‖G‖ ‖ξ ◦ (id +η1)− ξ ◦ (id +η2)‖ ≤ ‖G‖ ‖Dξ‖ ‖η1 − η2‖. Using the definitions of ξ and the properties of the bump function ρ, we have that ‖Dξ‖ ≤ sup |x|≤r ‖DX(x)−A‖+ 4 r sup |x|≤r |X(x)−Ax|. 6 Linearization via the Lie Derivative By the continuity of DX, there is some positive number r such that ‖DX(x)− A‖ < 1/(10‖G‖) whenever |x| ≤ r. By Taylor’s theorem (applied to the C1 function X) and the obvious estimate of the integral form of the remainder, if |x| ≤ r, then |X(x) − Ax| < r/(10‖G‖). For the number r > 0 just chosen, we have the estimate ‖G‖‖Dξ‖ < 1/2; and therefore, F is a contraction on B. By the contraction mapping theorem applied to the restriction of F on the closed subspace B0 of B consisting of those elements that vanish at the origin, the equation (2.7) has a unique solution η ∈ B0, which also satisfies the infinitesimal conjugacy equation (2.3). We will show that h := id +η is a local conjugacy. To do this recall the following elementary fact about Lie differentiation: If U , V , and W are vector fields, φt is the flow of U , and LUV = W , then d dt Dφ−t(φt(x))V (φt(x)) = Dφ−t(φt(x))W (φt(x)). Apply this result to the infinitesimal conjugacy equation (2.3) to obtain the identity d dt (e−tAη(etAx)) = e−tAξ(h(etAx)). Using the definitions of h and Y , it follows immediately that d dt (e−tAh(etAx)) = −e−tAAh(etAx) + e−tAY (h(etAx)) and (by the product rule) e−tA d dt h(etAx) = e−tAY (h(etAx)). Therefore, the function given by t 7→ h(etAx) is the integral curve of Y starting at the point h(x). But, by the definition of the flow Yt of Y , this integral curve is the function t 7→ Yt(h(x)). By uniqueness, h(etAx) = Yt(h(x)). Because Y is linear on the complement of a compact set, Gronwall’s inequality can be used to show that the flow of Y is complete. Hence, the conjugacy equation holds for all t ∈ R. It remains to show that the continuous function h : Rn → Rn given by h(x) = x + η(x) is a homeomorphism. Since η is bounded on Rn, the map h = id +η is surjective. To see this, choose y ∈ Rn, note that the equation h(x) = y has a solution of the form x = y + z if z = −η(y + z), and apply Brouwer’s fixed point theorem to the map z 7→ −η(y + z) on the ball of radius ‖η‖ centered at the origin. (Using this idea, it is also easy to prove that h is proper; that is, the inverse image under h of every compact subset of Rn is compact.) We will show that h is injective. If x and y are in Rn and h(x) = h(y), then Yt(h(x)) = Yt(h(y)) and, by the conjugacy relation, Atx+ η(Atx) = Aty+ η(Aty). By the linearity of At, we have that |At(x− y)| = |η(Aty)− η(Atx)|. (2.8) Carmen Chicone & Richard Swanson 7 For each nonzero u in Rn, the function t 7→ Atu = etAu is unbounded on R. Hence, either x = y or the left side of equation (2.8) is unbounded for t ∈ R. Since η is bounded, x = y; and therefore, the map h is injective. By Brouwer’s theorem on invariance of domain, the bijective continuous map h is a homeomorphism. (Brouwer’s theorem can be avoided by using instead the following elementary fact: A continuous, proper, bijective map from Rn to Rn is a homeomorphism.) 3 Smooth Conjugacy In the classic paper [H60M], Hartman shows that if a > b > 0 and c 6= 0, then there is no C1 linearizing conjugacy at the origin for the analytic differential equation ẋ = ax, ẏ = (a− b)y + cxz, ż = −bz. (3.1) On the other hand, he proved the following two important results. (1) If a C2 vector field has a rest point such that either all eigenvalues of its linearization have negative real parts or all eigenvalues have positive real parts, then the vector field is locally C1 conjugate to its linearization. (2) If a C2 planar vector field has a hyperbolic rest point, then the vector field is locally C1 conjugate to its linearization. Hartman proves the analogs of these theorems for maps and then derives the corresponding theorems for vector fields as corollaries. We will work directly with vector fields and thereby use standard methods from the theory of ordinary differential equations to obtain these results. We also note that S. Sternberg proved that the analytic planar system ẋ = −x, ẏ = −2y + x2 (3.2) is not C2 linearizable. Hence, it should be clear that the proofs of Hartman’s results on the existence of (maximally) smooth linearizations will require some delicate estimates. Nevertheless, as we will soon see, the strategy used in these proofs is easy to understand. Although the starting point for the proof of Theorem 2.1, namely, the dif- ferentiation with respect to t of the desired conjugacy relation (2.1) and the inversion of the operator LA as in display (2.6), leads to the simple proof of the existence of a conjugating homeomorphism given in Section 2, it turns out this strategy does not produce smooth conjugaces. This fact is illustrated by linearizing the scalar vector field given by X(x) = −ax+f(x) where a > 0. Sup- pose that f vanishes outside a sufficiently small open subset of the origin with radius r > 0 so that h(x) = x+η(x) is the continuous linearizing transformation where η(x) = ∫ ∞ 0 e−atf(eatx+ η(eatx)) dt 8 Linearization via the Lie Derivative as in the proof of Theorem 2.1. With F := f ◦ (id +η), u := eat, and x 6= 0, the function η is given by η(x) = 1 a ∫ r/|x| 1 F (ux) u2 du. Moreover, if x > 0, then (with w = ux) η(x) = x a ∫ r x F (w) w2 dw, and if x < 0, then η(x) = −x a ∫ x −r F (w) w2 dw. If η were continuously differentiable in a neighborhood of the origin, then we would have the identity η′(x) = 1 a ∫ r x F (w) w2 dw − F (x) ax for x > 0 and the identity η′(x) = −1 a ∫ x −r F (w) w2 dw − F (x) ax for x < 0. Because the left-hand and right-hand derivatives agree at x = 0, it would follow that ∫ r −r F (w) w2 dw = 0. But this equality is not true in general. For example, it is not true if f(x) = ρ(x)x2 where ρ is a bump function as in the proof of Theorem 2.1. In this case, the integrand is nonnegative and not identically zero. There are at least two ways to avoid the difficulty just described. First, note that the operator LA, for the case Ax = −ax, is formally inverted by running time forward instead of backward. This leads to the formal inverse given by (Gη)(x) := − ∫ ∞ 0 eatη(e−atx) dt and the fixed point equation η(x) = − ∫ ∞ 0 eatf(e−atx+ η(e−atx)) dt. In this case, no inconsistency arises from the assumption that η′(0) exists. In fact, in the last chapter of this paper, we will show that this method does Carmen Chicone & Richard Swanson 9 produce a smooth conjucacy for certain “special vector fields”, for example, the scalar vector fields under consideration here (see Theorem 3.8). Another idea that can be used to avoid the difficulty with smoothness is to differentiate both sides of the conjugacy relation etAh(x) = h(Xt(x)) (3.3) with respect to t, or equivalently for the scalar differential equation, to use the change of coordinates u = x + η(x). With this starting point, it is easy to see that η determines a linearizing transformation if it is a solution of the first order partial differential equation Dη(x)X(x) + aη(x) = −f(x). To solve it, replace x by the integral curve t 7→ φt(x) where φt denotes the flow of X, and note that (along this characteristic curve) d dt η(φt(x)) + aη(φt(x)) = −f(φt(x)). By variation of constants, we have the identity d dt eatη(φt(x)) = −eatf(φt(x)), and (after integration on the interval [0, t]) it follows that the function η given by η(x) = ∫ ∞ 0 eatf(φt(x)) dt (3.4) determines a linearizing transformation h = id +η if the improper integral con- verges on some open interval containing the origin. The required convergence is not obvious in general because the integrand of this integral contains the ex- ponential growth factor eat. In fact, to prove that η is continuous, a uniform estimate is required for the growth rate of the family of functions t 7→ |f(φt(x))|, and to show that η is continuously differentiable, a uniform growth rate estimate is required for their derivatives. The required estimates will be obtained in the next section where we will show that η is smooth for a hyperbolic sinks. For the scalar case as in equation (3.4), f(x) is less than a constant times x2 near the origin, and the solution φt(x) is approaching the origin like e−atx. Because this quantity is squared by the function f , the integral converges. To test the validity of this method, consider the example ẋ = −ax+x2 where the flow can be computed explicitly and the integral (3.4) can be evaluated to obtain the smooth near-identity linearizing transformation h : (−a, a) → R given by h(x) = x+ x2 a− x . 10 Linearization via the Lie Derivative 3.1 Hyperbolic Sinks The main result of this section is the following theorem. Theorem 3.1 (Hartman). Let X be a C2 vector field on Rn such that X(0) = 0. If every eigenvalue of DX(0) has negative real part, then X is locally C1 conjugate to its linearization at the origin. The full strength of the natural hypothesis that X is C2 is not used in the proof; rather, we will use only the weaker hypothesis that X is C1 and certain of its partial derivatives are Hölder on some fixed neighborhood of the origin. A function h is Hölder on a subset U of its domain if there is some (Hölder exponent) µ with 0 < µ ≤ 1 and some constant M > 0 such that |h(x)− h(y)| ≤M |x− y|µ whenever x and y are in U . In the special case where µ = 1, the function h is also called Lipschitz. As a convenient notation, let C1,µ denote the class of C1 functions whose first partial derivatives are all Hölder with Hölder exponent µ. Recall the definition of Hölder spectral exponents given in Section 1. We will prove the following generalization of Theorem 3.1. Theorem 3.2 (Hartman). Let X be a C1,1 vector field on Rn such that X(0) = 0. If every eigenvalue of DX(0) has negative real part and µ > 0 is smaller than the Hölder spectral exponent, then there is a near-identity C1,µ-diffeomorphism defined on some neighborhood of the origin that conjugates X to its linearization at the origin. The strategy for the proof of Theorem 3.2 is simple; in fact, the proof is by a finite induction. By a linear change of coordinates, the linear part of the vector field at the origin is transformed to a real Jordan canonical form where the diagonal blocks are ordered according to the real parts of the corresponding eigenvalues, and the vector field is decomposed into (vector) components cor- responding to these blocks. A theorem from invariant manifold theory is used to “flatten” the invariant manifold corresponding to the block whose eigenval- ues have the largest real part onto the corresponding linear subspace. This transforms the original vector field into a special form which is then “partially linearized” by a near-identity diffeomorphism; that is, the flattened—but still nonlinear—component of the vector field is linearized by the transformation. This reduces the dimension of the linearization problem by the dimension of the flattened manifold. The process is continued until the system is completely linearized. Finally, the inverse of the linear transformation to Jordan form is applied to return to the original coordinates so that the composition of all the coordinate transformations is a near-identity map. We will show that the nonlinear part of each near-identity partially lineariz- ing transformation is given explicitly by an integral transform∫ ∞ 0 e−tBg(ϕt(x)) dt Carmen Chicone & Richard Swanson 11 where g is given by the nonlinear terms of the component function of the vector field corresponding to the linear block B and ϕt is the nonlinear flow. The technical part of the proof is to demonstrate that these transformations maintain the required smoothness. This is done by repeated applications of Lemma 2.2 to prove that “differentiation under the integral sign” is permitted. Because maximal smoothness is obtained, it is perhaps not surprising that some of the estimates required to majorize the integrand of the integral transform are rather delicate. In fact, the main difficulty is to prove that the exponential rate of decay toward zero of the functions t 7→ g(ϕt(x)) and t 7→ gx(ϕt(x)), defined on some open neighborhood of x = 0, is faster than the exponential rate at which the linear flow etB moves points away from the origin in reverse time. As in Section 2, the original vector field X can be expressed in the “almost linear” form X(x) = Ax+ (X(x)−Ax). There is a linear change of coordinates in Rn such that X, in the new coordinates, is the almost linear vector field Y (x) = Bx+ (Y (x)−Bx) where the matrix B is in real Jordan canonical form with diagonal blocks B1 and B2, every eigenvalue of B2 has the same negative real part −b2, and every eigenvalue of B1 has its real part strictly smaller than −b2. The corresponding ODE has the form ẋ1 = B1 x1 + P1(x1, x2), ẋ2 = B2 x1 + P2(x1, x2) (3.5) where x = (x1, x2) and (P1(x1, x2), P2(x1, x2)) = (Y (x)− Bx). Let c be a real number such that −b < −c < 0, and note that if the augmented system ẋ1 = B1 x1 + P1(x1, x2), ẋ2 = B2 x1 + P2(x1, x2), ẋ3 = −cx3 is linearized by a near-identity transformation of the form u1 = x1 + α1(x1, x2, x3), u2 = x2 + α2(x1, x2, x3), u3 = x3, then the ODE (3.5) is linearized by the transformation u1 = x1 + α1(x1, x2, 0), u2 = x2 + α2(x1, x2, 0). More generally, let C1,L,µ denote the class of all systems of the form ẋ = Ax+ f(x, y, z), ẏ = By + g(x, y, z), ż = Cz (3.6) where x ∈ Rk, y ∈ R`, and z ∈ Rm; where A, B, and C are square matrices of the corresponding dimensions; B is in real Jordan canonical form; 12 Linearization via the Lie Derivative • every eigenvalue of B has real part −b < 0; • every eigenvalue of A has real part less than −b; • every eigenvalue of C has real part in an interval [−c,−d] where −b < −c and −d < 0; and F : (x, y, z) 7→ (f(x, y, z), g(x, y, z)) is a C1 function defined in a bounded product neighborhood Ω = Ωxy × Ωz (3.7) of the origin in (Rk × R`)× Rm such that • F (0, 0, 0) = 0 and DF (0, 0, 0) = 0, • the partial derivatives Fx and Fy are Lipschitz in Ω, and • the partial derivative Fz is Lipschitz in Ωxy uniformly with respect to z ∈ Ωz and Hölder in Ωz uniformly with respect to (x, y) ∈ Ωxy with Hölder exponent µ. System (3.6) satisfies the (1, µ) spectral gap condition if (1 + µ)c < b. We will show that system (3.6) can be linearized by a C1 near-identity trans- formation of the form u = x+ α(x, y, z), v = y + β(x, y, z), w = z. (3.8) The proof of this result is given in three main steps: an invariant manifold theorem for a system with a spectral gap is used to find a preliminary near- identity C1 map, as in display (3.8), that transforms system (3.6) into a system of the same form but with the new function F = (f, g) “flattened” along the coordinate subspace corresponding to the invariant manifold. Next, for the main part of the proof, a second near-identity transformation of the same form is constructed that transforms the flattened system to the partially linearized form ẋ = Ax+ p(x, y, z), ẏ = By, ż = Cz (3.9) where A, B, and C are the matrices in system (3.6) and the function p has the following properties: • p is C1 on an open neighborhood Ω = Ωx×Ωyz of the origin in Rk× (R`× Rm); Carmen Chicone & Richard Swanson 13 • p(0, 0, 0) = 0 and Dp(0, 0, 0) = 0; • The partial derivative px is Lipschitz in Ω; • The partial derivatives py and pz are Lipschitz in Ωx uniformly with re- spect to (y, z) ∈ Ωyz and Hölder in Ωyz uniformly with respect to x ∈ Ωx. The final step of the proof consists of three observations: The composition of C1 near-identity transformations of the form considered here is again a C1 near- identity transformation; the dimension of the “unlinearized” part of the system is made strictly smaller after applying the partially linearizing transformation, and the argument can be repeated as long as the system is not linearized. In other words, the proof is completed by a finite induction. The required version of the invariant manifold theorem is a special case of a more general theorem (see, for example, Yu. Latushkin and B. Layton [LL99]). For completeness, we will formulate and prove this special case. Our proof can be modified to obtain the general result. For notational convenience, let us view system (3.6) in the compact form Ẋ = AX + F (X , z), ż = Cz (3.10) where X = (x, y), A = ( A 0 0 B ) , and F := (f, g). Hyperbolic estimates for the corresponding linearized equations are used repeatedly. In particular, in view of the hypotheses about the eigenvalues of A, B, and C, it follows that if ε > 0 and 0 < λ < d, then there is a constant K > 1 such that ‖etA‖ ≤ Ke−(b−ε)t, ‖etC‖ ≤ Ke−λt ‖e−tC‖ ≤ Ke(c+ε)t (3.11) for all t ≥ 0. Theorem 3.3. If the (1, µ) spectral gap condition holds for system (3.10), then there is an open set Ωz ⊂ Rm containing z = 0 and a C1,µ function γ : Ωz → Rk+` such that γ(0) = Dγ(0) = 0 whose graph (the set {(X , z) ∈ Rk+` × Rm : X = γ(z)}) is forward invariant. As a remark, we mention that the smoothness of γ cannot in general be improved by simply requiring additional smoothness of the vector field. Rather, the smoothness of the invariant manifold can be improved only if additional requirements are made on the smoothness of the vector field and on the length of the spectral gap (see [LL99] and the references therein). For these reasons, it seems that the technical burden imposed by working with Hölder functions cannot be avoided by simply requiring additional smoothness of the vector field unless additional hypotheses are made on the eigenvalues of the linearization at 14 Linearization via the Lie Derivative the origin as well. Also, we mention that our proof illustrates the full power of the fiber contraction principle introduced by M. Hirsch and C. Pugh in [HP70] as a method for proving the smoothness of functions obtained as fixed points of contractions. To describe the fiber contraction method in our setting, let us consider a metric subspace D of a Banach space of continuous functions defined on Ω ⊂ Rm with values in Rp, and let us suppose that Γ : D → D is a contraction (on the complete metric space D) with fixed point γ. (In the analysis to follow, Γ is given by an integral transform operator.) We wish to show that γ is differentiable. Naturally, we start by formally differentiating both sides of the identity η(z) = Γ(η)(z) with respect to z to obtain the identity Dγ(z) = ∆(γ,Dγ)(z) where the map Φ 7→ ∆(γ,Φ) is a linear operator on a metric—not necessarily complete— subspace J of continuous functions from Ω to the bounded linear maps from Rm to Rp. We expect the derivative Dη, if it exists, to satisfy the equation Φ = ∆(η,Φ). Hence, J is a space of “candidates for the derivative of γ”. The next step is to show that the bundle map Λ : D × J → D × J defined by Λ(γ,Φ) = (Γ(γ),∆(γ,Φ)) is a fiber contraction; that is, for each γ ∈ D, the map Φ → ∆(γ,Φ) is a contraction on J with respect to a contraction constant that does not depend on the choice of γ ∈ D. The fiber contraction theorem (see [HP70] or, for more details, [C99]) states that if γ is the globally attracting fixed point of Γ and if Φ is a fixed point of the map Φ → ∆(γ,Φ), then (γ,Φ) is the globally attracting fixed point of Λ. The fiber contraction theorem does not require J to be a complete metric space. This leaves open the possibility to prove the existence of a fixed point in the fiber over γ by using, for example, Schauder’s theorem. But, for our applications, the space J will be chosen to be complete so that the existence of the fixed point Φ follows from an application of the contraction mapping theorem. After we show that Λ is a fiber contraction, the following argument can often be used to prove the desired equality Φ = Dγ. Find a point (γ0,Φ0) ∈ D × J such that Dγ0 = Φ0, define a sequence {(γj ,Φj)}∞j=0 in D × J by γj = Γ(γj−1), Φj = ∆(γj−1,Φj−1), and prove by induction that Dγj = Φj for every positive integer j. By the fiber contraction theorem, the sequence {γj}∞j=0 converges to γ and the sequence {Dγj}∞j=0 converges to the fixed point Φ of the map Φ → ∆(γ,Φ). If the con- vergence is uniform, then by a standard theorem from advanced calculus—the uniform limit of a sequence of differentiable functions is differentiable and equal to the limit of the derivatives of the functions in the sequence provided that the sequence of derivatives is uniformly convergent—the function γ is differentiable and its derivative is Φ. Let us prove Theorem 3.3. Carmen Chicone & Richard Swanson 15 Proof. The graph of γ is forward invariant if and only if Ẋ = Dγ(z)ż whenever X = γ(z). Equivalently, the identity Dγ(z)Cz −Aγ(z) = F (γ(z), z) (3.12) holds for all z in the domain of γ. The function γ will satisfy identity (3.12) if γ is C1 and d dτ e−τAγ(eτCz) = e−τAF (γ(eτCz), eτCz). In this case, by integration on the interval [−τ, 0] followed by a change of vari- ables in the integral on the right-hand side of the resulting equation, it follows that γ(z)− eτAγ(e−τCz) = ∫ τ 0 etAF (γ(e−tCz), e−tCz) dt. If γ is a C1 function such that γ(0) = Dγ(0) = 0 and limτ→∞|eτAγ(e−τCz)| = 0, then the graph of γ will be (forward) invariant provided that γ(z) = Γ(γ)(z) := ∫ ∞ 0 etAF (γ(e−tCz), e−tCz) dt. (3.13) For technical reasons, it is convenient to assume that γ is defined on all of Rm and that F is “cut off” as in the proof of Theorem 2.1 to have support in an open ball at the origin in Rk+`+m of radius r > 0 so that the new function, still denoted by the symbol F , is defined globally with ‖DF‖ bounded by a small number ρ > 0 to be determined. Recall that both r and ρ can be chosen to be as small as we wish. This procedure maintains the smoothness of the original function and the modified function agrees with the original function on some open ball centered at the origin and with radius r0 < r. Also, the graph of the restriction of a function γ that satisfies the equation (3.13) to an open subset of Ωz, containing the origin and inside the ball of radius r0, is forward invariant for system (3.10) because the the modified differential equation agrees with the original differential equation in the ball of radius r0. We will show that there is a solution γ of equation (3.13) such that γ ∈ C1,µ and γ(0) = Dγ(0) = 0. Let B denote the Banach space of continuous functions from Rm to Rk+` that are bounded with respect to the norm given by ‖γ‖B := sup { |γ(z)| |z| : z ∈ Rm \ {0} } . Also, note that convergence of a sequence in the B-norm implies uniform con- vergence of the sequence on compact subsets of Rm. Let D := {γ ∈ B : |γ(z1) − γ(z2)| ≤ |z1 − z2|}. Note that if γ ∈ D, then ‖γ‖B ≤ 1. We will also show that D is a closed subset, and hence a complete 16 Linearization via the Lie Derivative metric subspace, of B. In fact, if {γk}∞k=1 is a sequence in D that converges to γ in B, then |γ(z1)− γ(z2)| ≤ |γ(z1)− γk(z1)|+ |γk(z1)− γk(z2)|+ |γk(z2)− γ(z2)| ≤ ‖γ − γk‖B|z1|µ + |z1 − z2|+ ‖γ − γk‖B|z2|µ. (3.14) To see that γ ∈ D, pass to the limit as k →∞. We will show that if ρ is sufficiently small, then Γ, the operator defined in display (3.13), is a contraction on D. Recall the hyperbolic estimates 3.11, choose ε so small that c − b + 2ε < 0, let γ ∈ D, and note that |Γ(γ)(z1)− Γ(γ)(z2)| ≤ ∫ ∞ 0 Ke−(b−ε)t‖DF‖(|γ(e−tCz1)− γ(e−tCz2)| + |e−tC(z1 − z2)|) dt (3.15) ≤ ∫ ∞ 0 Ke−(b−ε)tρ(2|e−tC(z1 − z2)|) dt ≤ ( 2Kρ ∫ ∞ 0 e(c−b+2ε)t dt ) |z1 − z2|. (3.16) By taking ρ sufficiently small, it follows that |Γ(γ)(z1)− Γ(γ)(z2)| ≤ |z1 − z2|. Also, using similar estimates, it is easy to show that |Γ(γ1)(z)− Γ(γ2)(z)| ≤ K2ρ b− c− 2ε ‖γ1 − γ2‖B|z|. Hence, if ρ = ‖DF‖ is sufficiently small, then Γ is a contraction on the complete metric space D; and therefore, it has a unique fixed point γ∞ ∈ D. We will use the fiber contraction principle to show that γ∞ ∈ C1,µ. Before modification by the bump function, there is some open neighborhood of the origin on which FX is Lipschitz, Fz is Lipschitz in X , and Fz is Hölder in z. Using this fact and the construction of the bump function, it is not difficult to show that there is a constant M̄ > 0 such that (for the modified function F ) |Fz(X1, z1)− Fz(X2, z2)| ≤ M̄(|X1 −X2|µ + |z1 − z2|µ) (3.17) and |FX (X1, z1)− FX (X2, z2)| ≤ M̄(|X1 −X2|µ + |z1 − z2|µ). (3.18) Moreover, M̄ is independent of r as long as r > 0 is smaller than some preas- signed positive number. In view of the (1, µ) spectral gap condition, −b + ε + (1 + µ)(c + ε) < 0 whenever ε > 0 is sufficiently small. For ε in this class, let K̄ := K3 ∫ ∞ 0 e(−b+ε+(1+µ)(c+ε))t dt.