À È Ê ÇÄÁ Ë ËÌ ÅË Problem: A class of dynamical systems characterized by a fast divergence of the orbits A paradigmatic example: the Arnold cat. The closure of a homoclinic orbit. The shadowing lemma. some aplications of the shadowing lemma.
8.1 General setting Consider a compact differentiable manifold G and a diffeomorphism φ : G G. Assume there is a metric on G. Definition 8.1: A closed subset Λ G is said to be hyperbolic if it is invariant under φ and moreover for every x Λ there is a decomposition of the tangent space T x G = E x (s) E x (u) with dime (s) > 0 and dime (u) > 0, smooth in x and satisfying the following properties: (i) The decomposition is invariant for the tangent map dφ(x) : T x G T φ(x) G, i.e., dφ(x)e x (s) E (s) φ(x), dφ(x)e(u) x E (u) φ(x). (ii) E x (s) is a contracting subspace and E x (u) is an expanding subspace; i.e., there are positive constants C and µ such that for all > 0 one has dφ (x)ξ Ce µ ξ for ξ E x (s), (8.1) dφ (x)ξ Ce µ ξ for ξ E (u) x. Hyperbolic systems 230
Example 8.2: The doubling map of the circle and some generalizations. Strictly speaing, not a diffeomorphism (not invertible), but... exhibits the relevant characteristic of hyperbolic systems, i.e., the exponentially fast separation of orbits. The tangent space is one dimensional: only the expanding subspace. More generally: Pic an integer 2 Define the map as x f(x) mod1. This is still hyperbolic. May be further generalized to a non linear model (figure). A real, continuous differentiable function f : [0, 1] [0,] with integer 2; want f (x) > 1. Non differentiability at a finite set of points allowed. Define the map as x f(x) mod1. This is still hyperbolic. 3 0 0 1 mod1 1 0 0 1 Example 8.3: The baer transformation. The most elementary two-dimensional example is the baer transformation. Not even continuous, but differentiable almost everywhere. Expanding direction: the horizontal one. Contracting direction: the vertical one. Exhibits the main character of hyperbolic system: a combination of shrining/stretching action so that area is preserved. Hyperbolic systems 231
8.1.1 The Arnold cat A paradigmatic, classical model of hyperbolic system. The linear automorphism φ of T 2 ( ) ( )( ) x 1 1 x φ = (mod1). y 1 2 y Tangent space at every point: the plane R 2. Differential of the map: the matrix ( ) 1 1 dφ =. 1 2 Contracting subspace E x (s) and expanding subspace E x (u) at every point x generated by the eigenvectors ( ) ( ) 5 1 5 1 u (s) = 1,, u (u) =,1 2 2 (a) (c) (b) corresponding to the eigenvalues λ + = 3 5 2, λ = 3+ 5 2. Exercise 8.4: Show that: (i) The set Q of point with rational coordinates is invariant for the map, and all points x Q are periodic. (ii) There are infinitely many orbits which are asymptotic to the origin (0,0) either in the future or in the past. Among them there are infinitely many homoclinic (doubly asymptotic) orbits. (iii) The same properties holds true for every periodic orbit. Hyperbolic systems 232
Step 0 Step 1 Step 2 Step 3 Step 5 Step 8 Hyperbolic systems 233
8.1.2 The closure of a homoclinic orbit T x W (u) Proposition 8.5: Let x be an homoclinic point on a transversal intersection of the stable and unstable manifolds emanating from a hyperbolic fixed point x of a differentiable map φ. Consider the closure of the orbit through x, namely [ ] Λ = {x } φ x. Then Λ is an hyperbolic set. Z x* T x W (s) x Main argument: at every point x of the homoclinic orbit the expanding and contracting subspaces are tangent to the unstable and stable manifold, respectively. T Ψ(x) W (s) T Ψ(x) W (u) Ψ(x) Proof. Λ is invariant and has only x as accumulation point. For every x Λ the tangent space is the sum of two independent subspaces T x M = T x W (s) T x W (u). The subspaces are independent at x 0 by the transversality hypothesis between W (s) and W (u). Since W (s) and W (u) are invariant we have dφ(t x W (s) ) = T φ(x) W (s), dφ(t x W (u) ) = T φ(x) W (u). Must prove: (i) the splitting is smooth; (ii) the subspaces T φ (x)w (s) and T φ (x)w (u) are contracting and expanding, respectively. Hyperbolic systems 234
Smoothness: need to prove it only for the accumulation point x. E (u) W (u) T Ψ (x) W (u) Must prove: T φ (x)w (s) T x W (s) = E (s) and T φ (x)w (u) T x W (u) = E (u) for ±. z W (u) For + : W (s) is tangent to E (s) at x (by the stable manifold theorem) η In view of φ (x) x conclude T φ (x) E (s). For proceed by contradiction. Assume that T ψ (x)w (u) remains transversal to E (u) ; show that in such a case W (u) has a self intersection. x (u) x α η x (s) W (s) E (u) Introduce local coordinates in the neighbourhood of x taing E (s), E (u) as axes. For large enough denote by (x (s),x(u) ) the coordinates of φ (x). Since W (s) is tangent to E (s) at x, get x (u) = o(x (s) ). Since φ (x) tends exponentially to x for +, get x (s) < Ce σ for some positive C and σ. Let z T φ (x)w (u) ; write z = (x (s) +α η,x (u) +η) with some η and with a positive sequence α. Show that the claim T φ (x)w (u) E (u) is equivalent to the claim α 0. Hyperbolic systems 235
By contradiction: let α 0 be false. Then a subsequence satisfies α > δ > 0. Tae a neighbourhood U of x ; let W (u) be the connected component of W (u) U through φ (x). Any point ζ W (u) can be represented as ζ = ( (s) x +α η +f (η),x (u) +η ) with f (η) = o(η). W (u) emanatingfromx isrepresented intheneighbourhoodoftheoriginas ( h(ξ),ξ ) withh(ξ) = o(ξ). Show that there exists such that g(η) = 0 where g(η) = x (s) + α η + f (η) h(x (u) + η). Implies W (u) W (s). E (u) ξ x (u) x T Ψ (x)w (u) h(ξ) ζ η x (s) W (u) α η f (η) z ζ W (u) E (u) If is large and α > δ > 0 get g(η) = x (s) +α η +f (η) h(x (u) +η) = x (s) +α η +o(η). For large enough will have x (s) < δη/2. Yields g(+δ) > δ 2 η +o(η) > 0, g( δ) < δ 2 η +o(η) < 0. By continuity g(η) has a zero in the interval [ δ,δ], which is a self intersection point of W (u). Contradicts the nown fact that W (u) can not have a self-intersection. Hyperbolic systems 236
Prove that T x W (s) is contracting. Since φ (x) x for ± it is enough to consider K for K high enough, so that φ (x) U(x ), a neighbourhood of x. Let η T φ (x)w (s), and let η = η (s) +η (u) with η (s) E (s) and η (u) E (u). Clearly η (u) = α η (s) with some sequence α 0 for. Let, e.g., α < 1/2. In view of η > η (s) η (u) > (1 α ) η (s) get η (s) < 2 η and so also η (u) < 2α η. Then dφ(x )η = dφ(x )(η (s) +η (u) ) Ce σ η (s) +Ce σ η (u) 2C ( e σ + α e σ) η. Since the map φ is differentiable get also dφ(x) dφ(x ) D x x De σ for some positive D. Since α 0 for large enough there exist µ and D such that dφ(x)η De µ η. Q.E.D. Hyperbolic systems 237
8.2 The shadowing lemma Definition 8.6: A sequence {q } Z is said to be an ε pseudoorbit in case one has q φ(q 1 ) < ε for all Z. Definition 8.7: An orbit {p } Z is said to be a δ shadowing orbit for the ε pseudoorbit {q } Z in case one has p q < δ for all Z. Proposition 8.8: (Anosov, Bowen) Let Λ M be an hyperbolic set for the differentiable mapping φ. If δ > 0 is small enough then there is ε > 0 such that the following holds true: every ε pseudoorbit in Λ has a δ shadowing orbit in M. 8.2.1 Using the contraction principle Let {q } Z be an ε pseudooorbit. Loo for a sequence {x j } j Z such that {p j = q j +x j } j Z is an orbit and x j < δ. Want the sequence {x j } j Z to satisfy (8.2) q j+1 +x j+1 = φ(q j +x j ). Using the trivial identity φ(q j +x j ) = φ(q j )+dφ(q j )x j + [ φ(q j +x j ) φ(q j ) dφ(q j )x j ] rewrite the equation (8.2) as (8.3) x j+1 dφ(q j )x j = f j (x j ) where f j (x j ) = φ(q j ) q j+1 + [ φ(q j +x j ) φ(q j ) dφ(q j )x j ]. The left member of (8.3) is a linear expression; the right member is a small vector plus a quantity o( x j ). Hyperbolic systems 238
Consider the linear space of bounded sequences x = {x j } j Z with the norm x = sup j x j. The left side of equation x j+1 dφ(q j )x j = f j (x j ) taes the form (1 A)x, where A : X X is the linear operator defined as (8.4) y = Ax y j = dφ(q j 1 )x j 1. Write the right side as y = F(x), where y j = f j (x j ). Get the equation in X (8.5) (1 A)x = F(x), F(x) = {f j (x j )} j Z. Lemma 8.9: The operator 1 A is an isomorphism, and its inverse L = (1 A) 1 satisfies with the constants C and µ as in (8.1). Proof. Consider the equation L C 1+e µ 1 e µ (8.6) (1 A)x = g with nown g X. The space X splits into disjoint invariant subspaces X (s) = { X (s) j }j Z and X(u) = { X (u) } j definition of A through the differential of φ). For every 0 (see (8.4) in the definition of hyperbolic set) (8.7) A x (s) Ce µ x (s) for all x (s) X (s), A x (u) Ce µ x (u) for all x (u) X (u). Project the equation (1 A)x = g on X (s), X (u) ; get (8.8) (1 A)x (s) = g (s), (1 A)x (u) = g (u) j Z (by Hyperbolic systems 239
First equation, (1 A)x (s) = g (s). Establish the convergence of the formal solution x (s) = s 0A s g (s), In view of (8.7) get the convergent series A s g (s) C e µs g (s) = s 0 Second equation, (1 A)x (u) = g (u). Multiply both sides by A 1 s 0 Establish the convergence of the formal solution C 1 e µ g(s), A 1 (1 A)x (u) = (1 A 1 )x (u) = A 1 g (u). x (u) = A 1 s 0A s g (u) ; get (1 A)g (u) = x (u) Ce µ. 1 e µ g(u) Collect the estimates, which gives the wanted inequality. Q.E.D. Bac to the proof of the shadowing lemma: rewrite the equation (8.5) as an equation for a fixed point in X: (8.9) x = G(x) = LF(x). Hyperbolic systems 240
Lemma 8.10: Let B δ X be a ball of radius δ around x = 0. If δ is small enough and if ε satisfies, e.g., (8.10) ε < A µ δ, A µ = δ(1 e µ ) 2C(1+e µ ) then G(x) is a contraction in B δ. Proof. Recall: G(x) = LF(x) by (8.9) and that F(x) = {f j (x j )} j Z by (8.5). Prove that G(B δ ) B δ for δ small enough. φ is differentiable; hence F(x) = sup fj (x) j sup j ε+a δ δ φ(qj ) q j+1 +sup φ(qj +x j ) φ(q j ) dφ(q j )x j j Thus G(x) < C(1+e µ ) 1 e µ (ε+a δδ). Choose δ such that a δ < A µ and a corresponding ε satisfying (8.10) Get G(x) < δ, i.e., G(Bδ ) B δ, as claimed. Hyperbolic systems 241
Prove that G(x) G(x ) < c x x for every pair x, x B δ, with some c < 1. In view of f j (x j ) = φ(q j ) q j+1 + [ φ(q j +x j ) φ(q j ) dφ(q j )x j ] and of the differentiability of φ get fj (x j ) f j (x j ) = φ(qj +x j ) φ(q j +x j ) dφ(q j)(x j x j ) = o( xj x j ). For δ small enough get also F(x) F(x ) < aδ x x with a δ δ 0 0. By lemma 8.9, with the same δ get G(x) G(x ) = LF(x) LF(x ) < C 1+e µ 1 e µ F(x) F(x ) < a δ 2A µ x x < 1 2 x x. Completion of the proof of the shadowing lemma By lemma 8.10 the equation x = G(x) = LF(x) has a unique solution in B δ X. Choose ε which satisfies ε < A µ δ as in (8.10), and the claim of the shadowing lemma follows. Q.E.D. Hyperbolic systems 242
8.3 Applications of the shadowing lemma There are many... 8.3.1 Periodic orbits in the neighbourhood of a hyperbolic set An interesting result. Proposition 8.11: Let {q } Z be a periodic ε pseudoorbit of period N, i.e., q +N = q for all. Then the corresponding shadowing orbit is periodic of period N. Proof. The shadowing orbit {p } Z satisfies q p < δ for all, and is unique. By periodicity, q +N = q ; hence q +N p +N = q p +N < δ, i.e., the translated orbit {p +N } Z shadows {q } Z. Weconclude (byuniqueness) thatthattheorbit{p } Z coincides withthetranslatedorbit{p +N } Z. That is: p = p +N, hence the shadowing orbit is periodic. Q.E.D. If the hyperbolic set Λ is a closed subset of the phase space G then the periodic orbit needs not be a subset of Λ E.g., if Λ is the closure of a homoclinic orbit then every point of the periodic orbit lies in a neighbourhood of the corresponding point of the pseudoorbit. Hyperbolic systems 243
8.3.2 Dynamics in the neighbourhood of a homoclinic orbit All the examples in this section are concerned with a set Λ which is the closure of a homoclinic orbit, doubly asymptotic to a fixed point x. Let U ε (x ) be an ε neighbourhood of x. Let x be a homoclinic intersection of the stable and unstable manifolds emanating from x. There are positive j, such that φ t x U ε (x ) for t j and t. Construct a n slice ω n Ω(x) of the homoclinic orbit: pic j, such that φ j x U ε and φ x U ε ; set n = j + +1; let ω n = { φ j x,...,x,...,φ x } be the sequence of n consecutive points of the orbit Ω(x) between φ j x and φ x. Construct an ε pseudoorbit in Λ. The sequence (with an infinite sequence of x on both sides) {...,x,x,x, ω n,x,x,x,...} U ε (x*) x* W (u) W (s) φ 2 x φ 3 x φ 4 x φ 3 x φ 1 x φ 2 x x φx is an ε pseudoorbit. The homoclinic orbit Ω(x) is the unique orbit that shadows the pseudoorbit. Hyperbolic systems 244
Proposition 8.12: There are infinitely many distinct homoclinic orbits tending asymptotically to the hyperbolic fixed point x for t ±. Proof. Construct a finite sequence by concatenating any number o times the n slice ω n with any finite number of repetitions of x. E.g., tae the finite sequence { ωn,x,x, ω n,x,x,x, ω n } There are infinitely many similar sequences. Construct an ε pseudo orbit by complete the finite sequence above with an infinite sequence of x on both sides. There exists a unique shadowing orbit, which is clearly different from Ω(x) (it maes more than one cycle along the stable and unstable manifolds). Prove that the shadowing orbit is a homoclinic orbit. For t > T large enough all points of the shadowing orbit lie on a δ neighbourhood of x. All these points must lie on the local stable manifold W (s) (if not, then the map sends them away). Hence they must tend to x for t +. Apply the same argument, mutatis mutandis, for t. Q.E.D. Hyperbolic systems 245
Proposition 8.13: The fixed point x is a cluster point of periodic points. That is: in any neighbourhood of x there are infinitely many periodic points. Proof. Pic δ > 0 arbitrary; tae the corresponding ε. Let ω n be a n slice (remar: n will depend on ε). Construct a periodic pseudoorbit. construct a finite sequence by alternating a slice ω n with a finite sequence of x ; e.g., { ωn,x,x,x, ω n,x } construct the periodic pseudoorbit by repeating ad infinitum the finite sequence above. The shadowing orbit is periodic, and has a point in a δ neighbourhood of x Q.E.D. Produce further examples of periodic orbits. Hyperbolic systems 246
Proposition 8.14: Let Q = { } q 0,...,q n and Q = { q 0 m},...,q be periodic pseudoorbits, and let P = { } p0,...,p n and Q = { } q 0,...,q m be the corresponding periodic orbits. Then there is an orbit asymptotic to P for t and to P for t +. Hints for the proof. Construct a pseudoorbit (semicolons separate different sections of the infinite sequence) {...;q0,...,q n ;q 0,...,q n ;x,...,x ;q 0;...,q m;q 0,...,q m;... } i.e., an infinite repetition of q 0,...,q n on the left side and an infinite repetition of q 0 ;...,q m right side, separated by an arbitrary number of x. on the Prove that the corresponding pseudoorbit has the wanted asymptotic properties. Hyperbolic systems 247
8.3.3 Hyperbolic mappings Proposition 8.15: Let M be a compact measurable manifold with a hyperbolic measure preserving mapping φ. Then there is a dense set of periodic orbits for φ with arbitrarily long period. Proof. Must that in a δ neighbourhood of every point x M one may find a periodic point of period long enough. Let ε 0 correspond to δ/2 in the shadowing lemma; let ε = min(ε 0,δ/2). By Poincaré s recurrence theorem there is x B ε (x) such that φ N (x) B ε for some N. Construct the periodic pseudoorbit {q } Z by setting q 0 = x, q 1 = φ(x),...,q N 1 = φ N 1 (x), q N = x and q j+n = q j for j Z. Satisfies φ ( φ N 1 (x) ) q N = φ N (x) x < ε. In a δ/2 neighbourhood of {q } Z there s a periodic orbit {p } Z of φ; in particular p 0 x p 0 x + x x < δ, Q.E.D. Hyperbolic systems 248