8

HYPERBOLIC SYSTEMS

• 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 moreoverfor every x ∈ Λ there is a decomposition of the tangent space TxG = E(s)x ⊕ E(u)x with dimE(s) > 0 anddimE(u) > 0, smooth in x and satisfying the following properties:

(i) The decomposition is invariant for the tangent map dφ(x) : TxG → Tφ(x)G , i.e.,

dφ(x)E(s)x ⊂ E(s)φ(x) , dφ(x)E(u)x ⊂ E(u)φ(x) .

(ii) E(s)x is a contracting subspace and E

(u)x is an expanding subspace; i.e., there are positive constants C

and µ such that for all k > 0 one has

(8.1)

∥

∥dφk(x)ξ∥

∥ ≤ Ce−kµ‖ξ‖ for ξ ∈ E(s)x ,∥

∥dφ−k(x)ξ∥

∥ ≤ Ce−kµ‖ξ‖ for ξ ∈ E(u)x .

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Example 8.2: The doubling map of the circle and some gen-eralizations.

• Strictly speaking, 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 expandingsubspace.

• More generally:◦ Pick an integer k ≥ 2◦ Define the map as x → f(x) mod 1. This is stillhyperbolic.

• May be further generalized to a non linear model (figure).◦ A real, continuous differentiable function f : [0, 1] →[0, k] with integer k ≥ 2; want f ′(x) > 1.

◦ Non differentiability at a finite set of points allowed.◦ Define the map as x → f(x) mod 1. This is stillhyperbolic.

mod 1

0 10

0 1

1

0

3

Example 8.3: The baker transformation.• The most elementary two-dimensional example is the baker 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 shrinking/stretching action so thatarea is preserved.

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8.1.1 The Arnold cat

A paradigmatic, classical model of hyperbolic system.• The linear automorphism φ of T2

φ

(

xy

)

=

(

1 11 2

)(

xy

)

(mod1) .

◦ Tangent space at every point: the plane R2.◦ Differential of the map: the matrix

dφ =

(

1 11 2

)

.

◦ Contracting subspace E(s)x and expanding subspaceE

(u)x at every point x generated by the eigenvectors

u(s) =

(

−1,√5− 12

)

, u(u) =

(√5− 12

, 1

)

corresponding to the eigenvalues

λ+ =3−

√5

2, λ− =

3 +√5

2.

(b)

(a)

(c)

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.

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Step 0 Step 1 Step 2

Step 3 Step 5 Step 8

Hyperbolic systems – 233

8.1.2 The closure of a homoclinic orbit

Proposition 8.5: Let x be an homoclinic point on a transver-sal intersection of the stable and unstable manifolds emanatingfrom a hyperbolic fixed point x∗ of a differentiable map φ. Con-sider the closure of the orbit through x, namely

Λ = {x∗} ∪[

⋃

k∈Z

φkx]

.

Then Λ is an hyperbolic set.

• Main argument: at every point x of the homoclinic orbitthe expanding and contracting subspaces are tangent tothe unstable and stable manifold, respectively.

x

Ψ(x)

x*

TxW(u)

TxW(s)

TΨ(x)W(s)

TΨ(x)W(u)

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

TxM = TxW(s) ⊕ TxW (u) .

◦ The subspaces are independent at x0 by the transversality hypothesis between W (s) and W (u).◦ Since W (s) and W (u) are invariant we have

dφ(TxW(s)) = Tφ(x)W

(s) , dφ(TxW(u)) = Tφ(x)W

(u) .

• Must prove: (i) the splitting is smooth; (ii) the subspaces Tφk(x)W (s) and Tφk(x)W (u) are contractingand expanding, respectively.

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• Smoothness: need to prove it only for the accumulationpoint x∗.

◦ Must prove: Tφk(x)W (s) → Tx∗W (s) = E(s) andTφk(x)W

(u) → Tx∗W (u) = E(u) for k → ±∞.

• For k → +∞: W (s) is tangent to E(s) at x∗ (by thestable manifold theorem)

◦ In view of φk(x) → x∗ conclude Tφk(x) → E(s).• For k → −∞ proceed by contradiction.

◦ Assume that Tψk(x)W (u) remains transversal toE(u); show that in such a case W (u) has a self in-tersection.

x∗

W (s)

η

αkη

TΨk(x)W(u)

z

E(u)

W (u)

W (u)

E(u)

x(s)k

x(u)k

• Introduce local coordinates in the neighbourhood of x∗ taking E(s), E(u) as axes.◦ For k large enough denote by (x(s)k , x

(u)k ) the coordinates of φ

k(x).

◦ Since W (s) is tangent to E(s) at x∗, get x(u) = o(x(s)).◦ Since φk(x) tends exponentially to x∗ for k → +∞, get |x(s)k | < Ce−σk for some positive C and σ.◦ Let z ∈ Tφk(x)W (u); write z = (x(s) + αkη, x(u) + η) with some η and with a positive sequence αk .

• Show that the claim Tφk(x)W (u) → E(u) is equivalent to the claim αk → 0.

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◦ By contradiction: let αk → 0 be false. Then a sub-sequence satisfies αk > δ > 0.

◦ Take a neighbourhood U of x∗; let W (u)k be the con-nected component of W (u) ∩ U through φk(x).

◦ Any point ζ ∈ W (u)k can be represented as ζ =(

x(s)k + αkη + fk(η), x

(u)k + η

)

with fk(η) = o(η).

◦ W (u)∗ emanating from x∗ is represented in the neigh-bourhood of the origin as

(

h(ξ), ξ)

with h(ξ) = o(ξ).

◦ Show that there exists k such that g(η) = 0 whereg(η) = x

(s)k + αkη + fk(η) − h(x(u) + η). Implies

W(u)k ∩W

(s)k 6= ∅.

x∗

z

ζ

αkη

η

TΨk(x)W(u)

W (u)

ζ ′

W (u)

E(u)x(s)k

E(u)

x(u)k

ξ

fk(η)

h(ξ)

◦ If k is large and αk > δ > 0 get

g(η) = x(s)k + αkη + fk(η)− h(x(u) + η) = x

(s)k + αkη + o(η) .

◦ For k large enough will have x(s)k < δη/2. Yields

g(+δ) >δ

2η + o(η) > 0 , g(−δ) < −δ

2η + 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 known fact that W (u) can not have a self-intersection.

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• Prove that TxW (s) is contracting.◦ Since φk(x) → x∗ for k → ±∞ it is enough to consider |k| ≥ K for K high enough, so that φk(x) ∈

U(x∗), a neighbourhood of x∗.

◦ Let η ∈ Tφk(x)W (s), and let η = η(s) + η(u) with η(s) ∈ E(s) and η(u) ∈ E(u).

◦ Clearly η(u) = αkη(s) with some sequence αk → 0 for k → ∞. Let, e.g., |αk| < 1/2.◦ In view of ‖η‖ > ‖η(s)‖ − ‖η(u)‖ > (1− αk)‖η(s)‖ get ‖η(s)‖ < 2‖η‖ and so also ‖η(u)‖ < 2αk‖η‖ .◦ Then

∥

∥dφ(x∗)η∥

∥ =∥

∥dφ(x∗)(η(s) + η(u))∥

∥ ≤ Ce−σ‖η(s)‖+ Ceσ‖η(u)‖ ≤ 2C(

e−σ + |αk|eσ)

‖η‖ .

◦ Since the map φ is differentiable get also∥

∥dφ(x)− dφ(x∗)∥

∥ ≤ D‖x− x∗‖ ≤ De−σk

for some positive D.

◦ Since αk → 0 for k 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 {qk}k∈Z is said to be an ε–pseudoorbit in case one has∣

∣qk − φ(qk−1)∣

∣ < ε for all k ∈ Z .

Definition 8.7: An orbit {pk}k∈Z is said to be a δ–shadowing orbit for the ε–pseudoorbit {qk}k∈Z in caseone has

|pk − qk| < δ for all k ∈ Z .Proposition 8.8: (Anosov, Bowen) Let Λ ⊂ M be an hyperbolic set for the differentiable mapping φ. Ifδ > 0 is small enough then there is ε > 0 such that the following holds true: every ε–pseudoorbit in Λ has aδ–shadowing orbit in M .

8.2.1 Using the contraction principle

Let {qk}k∈Z be an ε–pseudooorbit.• Look for a sequence {xj}j∈Z such that {pj = qj + xj}j∈Z is an orbit and |xj| < δ.• Want the sequence {xj}j∈Z to satisfy

(8.2) qj+1 + xj+1 = φ(qj + xj) .

◦ Using the trivial identity φ(qj + xj) = φ(qj) + dφ(qj)xj +[

φ(qj + xj)− φ(qj)− dφ(qj)xj]

rewritethe equation (8.2) as

(8.3) xj+1 − dφ(qj)xj = fj(xj)where fj(xj) = φ(qj)− qj+1 +

[

φ(qj + xj)− φ(qj)− dφ(qj)xj]

.◦ The left member of (8.3) is a linear expression;◦ the right member is a small vector plus a quantity o(|xj |).

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• Consider the linear space of bounded sequences x = {xj}j∈Z with the norm ‖x‖ = supj |xj | .• The left side of equation xj+1 − dφ(qj)xj = fj(xj) takes the form (1− A)x, where A : X → X is the

linear operator defined as

(8.4) y = Ax ⇔ yj = dφ(qj−1)xj−1 .• Write the right side as y = F (x), where yj = fj(xj).• Get the equation in X

(8.5) (1− A)x = F (x) , F (x) = {fj(xj)}j∈Z .

Lemma 8.9: The operator 1− A is an isomorphism, and its inverse L = (1− A)−1 satisfies

‖L‖ ≤ C 1 + e−µ

1− e−µwith the constants C and µ as in (8.1).

Proof. Consider the equation

(8.6) (1− A)x = gwith known g ∈ X .

• The space X splits into disjoint invariant subspaces X(s) ={

X(s)j

}

j∈Zand X(u) =

{

X(u)j

}

j∈Z(by

definition of A through the differential of φ).• For every k ≥ 0 (see (8.4) in the definition of hyperbolic set)

(8.7)‖Akx(s)‖ ≤ Ce−µk‖x(s)‖ for all x(s) ∈ X(s) ,

‖A−kx(u)‖ ≤ Ce−µk‖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)

Hyperbolic systems – 239

• First equation, (1− A)x(s) = g(s).◦ Establish the convergence of the formal solution

x(s) =∑

s≥0

Asg(s) ,

◦ In view of (8.7) get the convergent series∑

s≥0

∥

∥Asg(s)

∥

∥ ≤ C∑

s≥0

e−µs‖g(s)‖ = C1− e−µ ‖g

(s)‖ ,

• Second equation, (1− A)x(u) = g(u).◦ Multiply both sides by −A−1

−A−1(1− A)x(u) = (1− A−1)x(u) = −A−1g(u) .◦ Establish the convergence of the formal solution

x(u) = −A−1∑

s≥0

A−sg(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.

• Back 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 Bδ ⊂ X be a ball of radius δ around x = 0. If δ is small enough and if ε satisfies, e.g.,

(8.10) ε < Aµδ , Aµ =δ(1− e−µ)2C(1 + e−µ)

then G(x) is a contraction in Bδ.

Proof. Recall: G(x) = LF (x) by (8.9) and that F (x) = {fj(xj)}j∈Z by (8.5).

• Prove that G(Bδ) ⊂ Bδ for δ small enough.◦ φ is differentiable; hence

∥

∥F (x)∥

∥ = supj

∣

∣fj(x)∣

∣

≤ supj

∣

∣φ(qj)− qj+1∣

∣+ supj

∣

∣φ(qj + xj)− φ(qj)− dφ(qj)xj∣

∣

≤ ε+ aδδ

◦ Thus∥

∥G(x)∥

∥ <C(1 + e−µ)

1− e−µ (ε+ aδδ) .

◦ Choose δ such that aδ < Aµ and a corresponding ε satisfying (8.10)◦ Get

∥

∥G(x)∥

∥ < δ, i.e., G(Bδ) ⊂ Bδ, as claimed.

Hyperbolic systems – 241

• Prove that ‖G(x)−G(x′)‖ < c‖x− x′‖ for every pair x, x′ ∈ Bδ, with some c < 1.◦ In view of fj(xj) = φ(qj)− qj+1 +

[

φ(qj + xj)− φ(qj)− dφ(qj)xj]

and of the differentiability of φget

∣

∣fj(xj)− fj(x′j)∣

∣ =∣

∣φ(qj + xj)− φ(qj + x′j)− dφ(qj)(xj − x′j)∣

∣ = o(|xj − x′j |) .

◦ For δ small enough get also∥

∥F (x)− F (x′)∥

∥ < aδ|x− x′| with aδ −→δ→0

0 .

◦ By lemma 8.9, with the same δ get∥

∥G(x)−G(x′)∥

∥ =∥

∥LF (x)− LF (x′)∥

∥

< C1 + e−µ

1− eµ∥

∥F (x)− F (x′)∥

∥ <aδ2Aµ

‖x− x′‖ < 12‖x− x′‖ .

Q.E.D.

• Completion of the proof of the shadowing lemma◦ By lemma 8.10 the equation x = G(x) = LF (x) has a unique solution in Bδ ⊂ X .◦ Choose ε which satisfies ε < Aµδ as in (8.10), and the claim of the shadowing lemma follows.

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 {qk}k∈Z be a periodic ε–pseudoorbit of period N , i.e., qk+N = qk for all k. Thenthe corresponding shadowing orbit is periodic of period N .

Proof.

• The shadowing orbit {pk}k∈Z satisfies |qk − pk| < δ for all k, and is unique.• By periodicity, qk+N = qk; hence

|qk+N − pk+N | = |qk − pk+N | < δ ,i.e., the translated orbit {pk+N}k∈Z shadows {qk}k∈Z .

• We conclude (by uniqueness) that that the orbit {pk}k∈Z coincides with the translated orbit {pk+N}k∈Z .• That is: pk = pk+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 asubset of Λ

◦ E.g., if Λ is the closure of a homoclinic orbit then every point of the periodic orbit lies in aneighbourhood of the corresponding point of the pseudoorbit.

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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, doublyasymptotic to a fixed point x∗.

• Let Uε(x∗) be an ε–neighbourhood of x∗. Let x be a homoclinic intersection of the stable and unstablemanifolds emanating from x∗.

• There are positive j, k such that φtx ∈ Uε(x∗) for t ≤ −j and t ≥ k.

• Construct a n–slice 〈ωn〉 ⊂ Ω(x) of the homo-clinic orbit:

◦ pick j, k such that φ−jx ∈ Uε and φkx ∈Uε ;

◦ set n = j + k + 1; let〈ωn〉 =

{

φ−jx, . . . , x, . . . , φkx}

be the sequence of n consecutive pointsof the orbit Ω(x) between φ−jx and φkx .

• Construct an ε–pseudoorbit in Λ.◦ The sequence (with an infinite sequenceof x∗ on both sides)

{. . . , x∗, x∗, x∗, 〈ωn〉, x∗, x∗, x∗, . . .}is an ε–pseudoorbit.

◦ The homoclinic orbit Ω(x) is the uniqueorbit that shadows the pseudoorbit.

x

Uε(x*)

x*

φ4xφ3x φ2x

φx

φ−1x

φ−3xφ−2x

W (u)

W (s)

Hyperbolic systems – 244

Proposition 8.12: There are infinitely many distinct homoclinic orbits tending asymptotically to thehyperbolic fixed point x∗ for t → ±∞.Proof.

• Construct a finite sequence by concatenating any number o times the n–slice 〈ωn〉 with any finitenumber of repetitions of x∗.

◦ E.g., take the finite sequence{

〈ωn〉, x∗, x∗, 〈ωn〉, x∗, x∗, x∗, 〈ωn〉}

◦ There are infinitely many similar sequences.

• Construct an ε–pseudo orbit by complete the finite sequence above with an infinite sequence of x∗ onboth sides.

• There exists a unique shadowing orbit, which is clearly different from Ω(x) (it makes more than onecycle 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. Pick δ > 0 arbitrary; take the corresponding ε.

• Let 〈ωn〉 be a n–slice (remark: n will depend on ε).

• Construct a periodic pseudoorbit.◦ construct a finite sequence by alternating a slice 〈ωn〉 with a finite sequence of x∗; e.g.,

{

〈ωn〉, x∗, x∗, x∗, 〈ωn〉, 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 ={

q0, . . . , qn}

and Q′ ={

q′0, . . . , q′m

}

be periodic pseudoorbits, and let P ={

p0, . . . , pn}

and Q′ ={

q′0, . . . , q′m

}

be the corresponding periodic orbits. Then there is an orbit asymptoticto P for t → −∞ and to P ′ for t → +∞.

Hints for the proof.• Construct a pseudoorbit (semicolons separate different sections of the infinite sequence)

{

. . . ; q0, . . . , qn; q0, . . . , qn; x∗, . . . , x∗; q′0; . . . , q

′m; q

′0, . . . , q

′m; . . .

}

i.e., an infinite repetition of q0, . . . , qn on the left side and an infinite repetition of q′0; . . . , q

′m on the

right side, separated by an arbitrary number of x∗.

• 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 map-ping 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 longenough.

• Let ε0 correspond to δ/2 in the shadowing lemma; let ε = min(ε0, δ/2).

• By Poincaré’s recurrence theorem there is x ∈ Bε(x) such that φN (x) ∈ Bε for some N .

• Construct the periodic pseudoorbit {qk}k∈Z by settingq0 = x, q1 = φ(x), . . . , qN−1 = φ

N−1(x), qN = x

and qj+N = qj for j ∈ Z.◦ Satisfies

∣

∣

∣φ(

φN−1(x))

− qN∣

∣

∣=∣

∣φN (x)− x∣

∣ < ε .

• In a δ/2 neighbourhood of {qk}k∈Z there s a periodic orbit {pk}k∈Z of φ;

• in particular|p0 − x| ≤ |p0 − x|+ |x− x| < δ ,

Q.E.D.

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