HADAMARD-PERRON THEOREM

Transcription

1 HADAMARD-PERRON THEOREM CARLANGELO LIVERANI. Invariant manifold of a fixed point He we will discss the simplest possible case in which the existence of invariant manifolds arises: the Hadamard-Perron theorem. Definition. Given a smooth map T : X X, X being a Riemannian manifold, and a fixed point p X (i.e. T p = p) we call (local) stable manifold (of size δ) a manifold W s (p) sch that W s (p) = {x B δ (x) X lim d(t n x, p) = 0}. n Analogosly, we will call (local) nstable manifold (of size δ) a manifold W (p) sch that W (p) = {x B δ (x) X lim d(t n x, p) = 0}. n It is qite clear that T W s (p) W s (p) and T W (p) W (p) (Problem ). Less clear is that these sets deserve the name manifold. Yet, if one thinks of the Arnold cat at the point zero (which is a fixed point) it is obvios that the stable and nstable manifolds at zero are jst segments in the stable and nstable direction, the next Theorem shows that this is a qite general sitation. Theorem. (Hadamard-Perron). Consider an invertible map T : U R R, T C (U, R ), sch that T 0 = 0 and λ 0 (.) D 0 T = 0 µ where 0 < µ < < λ. That is, the map T is hyperbolic at the fixed point 0. Then there exists stable and nstable manifolds at 0. Moreover, T 0 W s() (0) = E s() (0) where E s() (0) are the expanding and contracting sbspaces of D 0 T. Proof. We will deal explicitly only with the nstable manifold since the stable one can be treated exactly in the same way by considering T instead of T. Since the map is continosly differentiable for each ε > 0 we can choose δ > 0 so that, in a δ-neighborhood of zero, we can write (.) T (x) = D 0 T x + R(x) where R(x) ε x, D x R ε. Date: Janary, 007. Sometime we will write W s δ (p) when the size really matters. By B δ (x) we will always mean the open ball of radis δ centered at x. Notice that if D0 T has eigenvales 0 < µ < < λ then one can always perform a change of variables sch that (.) holds. Also I am assming real positive eigenvales jst for simplicity, complex eigenvales with R(µ) < < R(λ) wold as well. Also note that the dimension is rather irrelevant in the following, an extension to operators on Banach space wold hold almost verbatim.

2 CARLANGELO LIVERANI.0.. Existence a fixed point argment. The first step is to decide how to represent manifolds. In the present case, since we deal only with crves, it seems very reasonable to consider the set of crves Γ δ,c passing throgh zero and close to being horizontal, that is the differentiable fnctions γ : [ δ, δ] R of the form t γ(t) = (t) and sch that γ(0) = 0; (, 0) γ c. It is immediately clear that any smooth crve passing throgh zero and with tangent vector, at each point, in the cone C := {(a, b) R b a c}, can be associated to a niqe element of Γ δ,c, jst consider the part of the crve contained in the strip {(x, y) R x δ}. Moreover, if γ Γ δ,c then γ B δ (0), provided c /. Notice that it sffices to specify the fnction in order to identify niqely an element in Γ δ,c. It is then natral to stdy the evoltion of a crve throgh the change in the associated fnction. To this end let s investigate how the image of a crve in Γ δ,c nder T looks like. T γ(t) = ( λt + R (t, (t)) µ(t) + R (t, (t)) ) := ( α (t) β (t) At this point the problem is clearly that the image it is not expressed in the way we have chosen to represent crves, yet this is easily fixed. First of all, α (0) = β (0) = 0. Second, by choosing ε < λ, we have α (t) > 0, that is, α is invertible. In addition, α ([ δ, δ]) [ λδ + εδ, λδ εδ] [ δ, δ], provided ε λ. Hence, α ). is a well defined fnction from [ δ, δ] to itself. Finally, d dt β α (t) = β (α (t)) α (α (t)) µc + ε λ ε c where, again, we have chosen ε c(λ µ) +c. We can then consider the map T : Γ δ,c Γ δ,c defined by t (.3) T γ(t) := β α (t) which associates to a crve in Γ δ,c its image nder T written in the chosen representation. It is now natral to consider the set of fnctions B δ,c = { C ([ δ, δ]) (0) = 0, c} in the vector space Lip([ δ, δ]). 3 As we already noticed B δ,c is in one-one correspondence with Γ δ,c, we can ths consider the operator ˆT : Lip([ δ, δ]) Lip([ δ, δ]) defined by (.4) ˆT = β α From the above analysis follows that ˆT (B δ,c ) B δ,c and that ˆT determines niqely the image crve. The problem is then redced to stdying the map ˆT. The easiest, althogh probably not the most prodctive, point of view is to show that ˆT is a contraction in the sp norm. Note that this creates a little problem since C it is not closed in the sp norm (and not even Lip([ δ, δ]) is closed). Yet, the set Bδ,c = { 3 This are the Lipschitz fnctions on [ δ, δ], that is the fnctions sch that sp t,s [ δ, δ] (s) (t) t s <.

3 UNSTABLE MANIFOLDS 3 (s) (t) Lip([ δ, δ]) (0) = 0, sp t,s [ δ, δ] t s c} is closed (see Problem ). Ths B δ,c Bδ,c. This means that, if we can prove that the sp norm is contracting, then the fixed point will belong to Bδ,c and we will obtain only a Lipschitz crve. We will need a separate argment to prove that the crve is indeed smooth. Let s start to verify the contraction property. Notice that α (t) = λ t + λ R (α ths, given, B δ,c, by Lagrange Theorem (t), (α (t))), α (t) α (t) λ ζ R, (α (t) α (t), (α (t)) (α (t))) ε { α λ (t) α (t) + (α (t)) (α (t)) }. This implies immediately (.5) α (t) α (t) λ ε λ ε. On the other hand (.6) Moreover, β (t) β (t) µ (t) (t) + ζ R, (0, (t) (t)) (µ + ε). (.7) β (t) µ + ε. Collecting the estimates (.5,.6,.7) readily yields ˆT ˆT β α β α + β α β α { λ } ε [µ + ε] λ + (µ + ε) ε σ, for some σ (0, ), provided ε is chosen small enogh. Clearly, the above ineqality immediately implies that there exists a niqe element γ Γ γ,c sch that T γ = γ, this is the local nstable manifold of Reglarity a cone field. As already mentioned, a separate argment is needed to prove that γ is indeed a C crve. To prove this, one possibility cold be to redo the previos fixed point argment trying to prove contraction in CLip (the C fnctions with Lipschitz derivative); yet this wold reqire to increase the reglarity reqirements on T. A more geometrical, more instrctive and more inspiring approach is the following. Define the cone field C θ,h (x, ) := {ξ B h (x) (a, b) = ξ x; a 0; b a θ}, with cδ, θ cδ and h δ. By constrction B h (x) γ C cδ,h for each x γ. We will stdy the evoltion of sch a cone field on γ. For all ξ C θ,h (x, ), if (a, b) = ξ x and (α, β) = T ξ T x, it holds (α, β) = D x T (a, b) + O(C (a, b) ). Ths, setting (α, β ) = D x T (a, b) and = β α, one can compte β α µλ [c h + θ],

4 4 CARLANGELO LIVERANI for some constant c depending only on T and δ. Accordingly, if h c θ, for same appropriate constant c, and δ is small enogh, there exists σ (0, ) sch that B h (x) T C θ,h (x, ) C σθ,h (T x, ). Hence, if x γ, γ B σ n h(t n x) C cδ,σ n h(t n x, 0) and, since T n γ γ, (.8) γ B σn h(x) C σn c,σ n h(x, v n ) where (a, av n ) = D T n xt n (, 0), for some a R +. The estimate (.8) clearly implies (.9) γ (x) = (, lim n v n) which indeed exists (see Problem 3). There is an isse not completely addresses in or formlation of Hadamard- Perron theorem: the niqeness of the manifolds. 4 It is not hard to prove that W s() (p) are indeed niqe (see Problem 4). There is another point of view that can be adopted in the stdy of stable and nstable manifolds: to grow the manifolds. This is done by starting with a very short crve in Γ δ,c, e.g. γ 0 (t) = (t, 0) for t [λ n δ, λ n δ], and showing that the seqence γ n := T n γ 0 converges to a crve in the strip [ δ, δ], independent of γ 0. From a mathematical point of view, in the present case, it corresponds to spell ot explicitly the proof of the fixed point theorem. Nevertheless, it is a more sggestive point of view and it is more convenient when the hyperbolicity is non niform. For example consider the map 5. ( x (.0) T := y) then 0 is a fixed point of the map bt D 0 T = x sin x + y x sin x + y 0 is not hyperbolic, yet, de to the higher order terms, there exist stable and nstable manifolds (see Problems 6, 7, 8). Problems Show that, if p is a fixed point, then T W s (p) W s (p) and T W (p) W (p). Prove that the set Bδ,c in section is closed with respect to the sp norm = sp t [ δ,δ] (t). 3 Prove that the limit in (.9) is well defined. 4 Prove that, in the setting of Theorem., the nstable manifold is niqe. (Hint: This amonts to show that the set of points that are attracted to zero are exactly the manifolds constrcted in Theorem.. Use the local hyperbolicity to show that.) 5 Show that Theorem. holds assming only T C (U, U). 4 Namely the dobt may remain that a less reglar set satisfying Definition exists. 5 Some times this is called Lewowicz map

5 UNSTABLE MANIFOLDS 5 6 Consider the Lewowicz map (.0), show that, given the set of crves Γ δ,c := {γ : [ δ, δ] R γ(t) = (t, (t)); γ(0) = 0; (t) [c t, ct]}, it is possible to constrct the map T : Γ δ,c Γ δ(+c δ), c in analogy with (.3). 7 In the case of the previos problem show that for each γ i Γ δ,c holds d( T γ, T γ ) ( cδ)d(γ, γ ). 8 Show that for the Lewowicz map zero has a niqe nstable manifold. (Hint: grow the manifolds, that is, for each n > define δ n := ρ n. Show that one can choose ρ sch that δ n δ n (+c δ n ). according to Problem 6 it follows that T : Γ δn,c Γ δn,c. Moreover, d( T n γ, T n n γ ) ( cδ i )d(γ, γ ). i= Finally, show that, setting γ n (t) = (0, t) Γ δn,c, the seqence T n γ n is a Cachy seqence that converges in C 0 to a crve in Γ,c invariant nder T.) Carlangelo Liverani, Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 0033 Roma, Italy. address: liverani@mat.niroma.it