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Lecture Notes on

Introduction to

Continuous Dynamical Systems

Fall, 2012

Lee, Keonhee

Department of Mathematics

Chungnam National Univeristy

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Chap 0. Introduction

What is a dynamical system?A dynamical system is a state space with an action.

0.1. Definition State space : Xℝ, Banach spaces, topological spaces,measure spaces, smooth manifolds, symplectic manifolds...

0.2. Definition An action on is a continuous (or differentiable, or measure preserving) map × → s.t. ⑴ ℝ, ℤ, ℝ∪, or ℕ∪, ⑵ for ∈, ⑶ for ∈ and ∈.

0.3. Definition ℤ is said to be a discrete dynamical system on , andℝ is said to be a continuous dynamical system on .

0.4. Remarks Let be a topological space. Note that every homeomorphism → generates a discrete dynamical system ×ℤ→ given by for ∈ and ∈ℤ. Conversely, every discrete dynamical system on is characterized by the homeomorphism · → given by · . Consequently we do not distinguish between a homeomorphism (or diffeomorphism) on and the discrete dynamical system on generated by .

A main goal of the theory of dynamical system is to understand "what happens the long time behavior of the orbits of ?"

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0.5. Classification of dynamical systems: - according to the state spaces and actions -

State spaces Actions( Discrete / Continuous )

Topological Dynamics topological spaces continuous mapshomeomorphisms

Differentiable Dynamics differentiable manifoldssymplectic manifolds

differentiable mapsdiffeomorphismssymplectomorphisms

Ergodic Theory measure spaces measure preserving maps

Dynamical Equations ℝ

Banach spacesdifference equationsdifferential equationstime scale dynamicsfractional differential equations

Numerical DynamicsControl Theory

Statistical Dynamics⋮

Our subject: Differentiable Dynamics (or Differentiable Dynamics System) Diffeomorphisms and Vector Fields (or Flows)

The theory of diffrentiable dynamical systems was originated by Steve Smale (1967) in the following paper: "Differentiable dynamics systems, Bull. of the AMS, Vol 73 (1967), 747-817." He used many tools and techniques in differential topology to develop the theory of differentiable dynamical systems.

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Basic knowledge for Differentiable Dynamical Systems 1. Differentiable manifolds: -manifolds, Riemannian manifolds, embedded submanifolds, immersed manifolds, tangent spaces, tangent maps, -maps, exponential maps, ⋯

2. Differential equations: Existence theorem, Uniqueness theorem, Regularity theorem, …

3. Linear algebra: Eigenvalue, Jordan canonical form, …

4. Differentiable topology: -topology, transversality, vector bundle, vector fields, …

5. Banach spaces : Contraction mapping theorem, Implicit function theorem, …

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Topics I hope to cover include

Chap⒈ Vector fields on manifolds

Chap⒉ Dynamical vocabularies

Chap⒊ Hyperbolicity

Chap⒋ Chain components and homoclinic classes

Chap⒌ Linear Poincaré flows

Chap⒍ Fundamental theorems for flows

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Chapter 1. Vector Fields on Manifolds

1.1. Notations ∘ = a -manifold of dimension , ≥ ≥. ∘ For each ∈, = the tangent space of at ; it is an -dimensional vector space. ∘ =

∈ : the tangent bundle; it is a -manifold of dimension 2.

∘ For convenience, we assume is a ∞-manifold of dimension , and also we assume has a Riemannian structure.

1.2. DefinitionA -vector field on is a -map → such that for each ∈, ∈.

Note that if → is a vector field on , then ∘ is the identity map on , where → is the projection map.

q p

1.3. Notation = the set of all -vector fields on with the -topology. Then can be considered as a subspace of .

1.4. topologyMotivation. For any two ∞-maps ℝ→ℝ, we say that ⑴ and are --close ( ) if for all ∈ℝ. ⑵ and are --close ( ) if and ′′ for all ∈ℝ.

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⑶ and are --close ( ) if , ⋯ , for all ∈ℝ. ⇓ Generalize

The -topology on is the topology induced by the -metric , where and are ∞-manifolds.

1.5. DefinitionA -flow on is a -map ×ℝ→ such that ⑴ , ⑵ , ∈, ∈ℝ.

1.6. RemarksLet ×ℝ→ be a -flow. Then ⑴ for each ∈ℝ, the map → defined by is a -diffeomorphism; is called a transition map. ⑵ for each ∈, the map ․ → given by ․ is a -curve in through .

1.7. DefinitionLet × ℝ→ be a -flow on for each ∈. Then the set ∈ ℝ is called the orbit of through , and is denoted by .

1.8. DefinitionLet ∈ , and let ∈. An integral curve of through is a map → such that ⑴ , ⑵ ′ for all ∈ .

′

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1.9. Remarks ∘ A -vector field on can be considered as an ODE on given by

′ , ∈ ※

∘ An integral curve of through can be considered as a solution curve of ※ with initial condition .

1.10. Theorem (Local Flow Theorem) For any ∈ and ∈, there are a neighborhood of , ≻,and a -map × → such that (1) ∈; (2)

=, ∈ and ∈.

′

Proof. If we apply the - Existence Theorem (since X is continuous) - Uniqueness Theorem (since X is locally Lipschitz) - Regularity Theorem (since X is ) for ẋ, then we get the -map with the above properties (1) and (2).

1.11. DefinitionThe -map obtained in Theorem 1.10 is called a local flow of at . In fact, can be considered as a collection of integral curves (or local solutions) of on ; → is an integral curve of through ∈.

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=

, if ≤

, if

NOTE: According to Theorem 1.10, every -vector field ∈ generate a local -flow. The natural question is

“which vector fields ∈ generate a global flow on ?”

1.12. Theorem (Global Flow Theorem) If is a compact ∞-manifold, then every vector field ∈ generate a -flow ; that is ×ℝ→ is a -map such that (1) is a flow, (2)

for all ∈×ℝ.

Here is called the flow on generate by .

Proof. Step 1. For each point of , there is a unique complete integral curve ℝ→ of through ; Since is locally Lipschitz, for each

∈, there is a unique integral curve of through . Let be the maximal domain of , and suppose ≠, say

∞. Choose a sequence () in such that → . Since M is compact, the sequence converges to a point, say , in . By Theorem 1.10, there is a local flow × → of at .

Choose ∈ such that and ∈.

Define → by

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Then is an integral curve of through . By the uniqueness, we have . Since ⊂ , we arrive at a contradiction. Consequently, we have b=∞. Similarly we can show that a=∞.

Step 2. Define a map ×→ by , where is the unique complete integral curve of through ∈. Then we can see that the map is well–defined and

(1) for all ∈, (2)

′ ′

(3) In fact, Consider two maps ℝ→ given by and Then and are integral curves of such that

By the uniqueness of integral curves with same initial point, we have . This means that for all ∈ℝ.

Step 3. is .(i) There exists such that for any ∈ , the transition map → is . 「∵ By the local flow theorem, for any p∈M,

there is -local flow × → of at . Since is compact, there are ⋯∈ such that ⋯ covers .

Consider the corresponding -local flows × → of at , respectively, for i=1,2,⋯,n. By the uniqueness of integral curves, we have ∣

×

for i=1,2,⋯,n. In particular, for any ∈

, is on for i=1,2,⋯,n.

Put = min,,⋯,. Then for any ∈ , is on each , and so is on . 」

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(ii) For any ∈ℝ, is on . 「∵ For any ∈ ℝ, choose ∈ℕ s.t.

, where is given in the above step (i). Then

∘⋯∘

and each

is on . Hence is on . 」

(iii) For any ∈×ℝ, is on a neighborhood of . 「∵ By the local flow theorem,

there are a neighborhood , constant , and a local flow of at such that

× .

This means that is on × ⋯ ※ Then we have is on × : a neighborhoodof . In fact, for any ∈× ,

∘ ∘ is since and

× are . 」

Homework 1. Let be a vector field on ℝ with its principal part ℝ → ℝ

↦ :

that is, ℝ → ℝ ℝ

↦ .

Find the flow generated by .

(Sol.)

※⇔

⇒ The solution of (※) with initial condition is cossin

sincosHence the flow generated by is given by

ℝ×ℝ → ℝ

↦ cossin sincos

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•

2. Give an example to show that1) is not locally Lipschiz2) the integral curve of through a point ∈ is not unique.

(Sol.) : a vector field on given by

.⇒ is not locally Lipschiz at :

→∞ as →.

There are two integral curves of through .Let →ℝ be given by

for all

→ℝ be given by for all .Then ′

′

. ∴ and are integral curves of at ∈ℝ. (≠)

3. Find a -vector field on a noncompact manifold such that the integral curves of are not complete.

(Sol.) Let be a ∞vector field on ℝ given by Then the integral curve of through ∈ℝ is not complete :

⇔ ⇔

⇔

⇔

.

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Let ⇒

is a integral curve of through with ∞ ≠ℝ

『 ∵ ․ ․ ′

』

∴ is not complete.

Homework1. Let ×ℝ→ be a -flow. For each ∈ℝ, the map → defined

by , ∈, is a -diffeomorphism.

(Sol.) We say that → is a -diffeomorphism if is bijective, and and are . (i) is bijective : For any ∈, we have ∘ , and ∘ . Hence , and so is bijective. (ii) is : Since ×ℝ→ is , × ×t→M is also . Note that × is a ∞-manifold which is diffeomorphic to , and so the map →× defined by is a ∞-diffeomorphism. Consequently, the map ×∘ is .

×

×

(iii) Similarly, we can show that is . ∴ is a -diffeomorphism.

2. For each ∈, the map ℝ→ defined by is a -curve in through .

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(Sol.) We say that a map → is a -curve in through if , and is . (i) (ii) Since ×ℝ→ is , ×ℝ ×ℝ→ is also . Note that ×ℝ is ∞-manifold which is diffeomorphic to ℝ, and so the map ℝ→ ×ℝ defined by is a ∞-diffeomorphism. Consequently, the map ×ℝ∘ is .

ℝ

× ℝ

×ℝ

1.13. DefinitionIf ×ℝ→ is a -flow, then the vector field → defined by

is called the velocity vector field of . Note that X is .

1.14. ConclusionThe qualitative theory of smooth vector fields and smooth flows are one and the same subject. However it is sometimes easier to describe the theory in terms of vector fields; sometimes in terms of flows.

Chapter 2. Dynamical Vocabularies

2.1. NotationFor each ∈ , denotes the -flow generated by .

2.2. DefinitionFor any ∈ and ∈, (1) ∈ℝ is called the orbit of through ; (2) is said to be singular if (or );

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(3) is said to be periodic (or closed) if for some but ≠ for all ; In this case, is called a periodic orbit with period s; (4) is said to be regular if it is not singular not periodic.

2.3. DefinitionFor any ∈ and ∈, (1) ∈ → for →∞ is called the -limit set of ; (2) ∈ → for →∞ is called the -limit set of .

2.4. Theorem (Basic properties of limit sets)Let ∈ and ∈. Then (1) ; (2) ≠∅; (3) for any ∈; (4) is invariant, that is, for all ∈ℝ, where, ∈ for some ⊂; (5) is connected; (6) is closed.Clearly the properties above are also true for the -limit sets.

Proofs of (1) and (2) are clear. (3),(4),(6) : (Homework)Proof of (5): Suppose is not connected. Then there is a separation of . Hence we can choose open sets in such that ⊂ ∪, ∩≠∅, ∩≠∅, and ∩∅.Take ∈∩ and ∈∩. Then ∃→∞ such that →, and ∃→∞ such that →.

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Put ∪. Then ∩∅.Since is a -curve, we know that ∀∃ ∈.Since is compact, we may assume ∃→∞ → ∈.Then, ∈∩. (#)

2.5. RemarksIf is not compact, the properties (2) and (5) in Theorem 2.4 do not hold in general. For example, consider a flow in ℝ given by the following figure.

ℝ×ℝ →ℝ

↦

If , then ∈ℝ → for →∞.If

, ∪ : it is not connected.

『 ∵ ∀∈ or ∃→∞ → . 』

2.6. Example (Vector field on )Let ∈ℝ . Then it is a ∞-manifold of dim 2. Consider a vector field on given by →

↦

What is the flow of ? It is not easy to express the equation of explicitly, but we can draw the phase portrait of (orbits of ) geometrically using the properties of as follows.Let , and . Then and are singularities of .

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『 ∵ and . 』For any ∈ , we have the following properties: (i) and are orthogonal: In fact, we get ․ ․

∴∈.

(ii) the norm of : ∥∥

→ →

(iii) the direction of : is tangent to the meridian of through . In fact, the plane containing the meridian of through is

given by

Since ∈, the phase portrait of is given by the following figure:

∀∈

We do not know the equation of exactly, but we can draw the phase portrait of .

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2.7. RemarksLet → be a -diffeomorphism and ∈ . Then the map → defined by , is a

-vector field on . Moreover, if → is an integral curve of , then ∘ → is an integral curve of . In particular, takes orbits of onto orbits of . Hence, we have the following well-defined map →

↦

↓ ↓

Homework (1) If is a subgroup of ℝ, then is either dense or discrete. (2) Let → be a continuous and surjective map. If is dense in then is dense in . (3) Let ∈ℝ be irrational. For any ∈ℕ, there exists ∈ℤ such that ∈ . (4) Find a local diffeomorphism ℝ → ; that is, for any ∈ℝ, there are a neighborhoods of and of , respectively, such that → is a diffeomorphism.

2.8. Example (Rational and Irrational Flows on )Step 1. Consider a covering map ℝ→ given by

coscoscossin sin

t 1

0 1 s

Then ⅰ is a local diffeomorphism,

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ⅱ takes the horizontal lines in ℝ to the parallel of latitudes in ; the vertical lines in ℝ to the meradians in . ⅲ ×

ⅳ ⇔ mod mod

Step 2. For any ∈ℝ, consider a ∞-vector field on on ℝ given by ; principal part.

Put . Then by Remarks 2.7,

ⅰ is a well-defined ∞-vector field on , ⅱ The orbits of is the image of the orbits of (= the lines of slope in ℝ) under .

Step 3. We show that ⅰ every orbits of is closed (or periodic) if is rational, ⅱ every orbits of is dense in if is irrational.

c+n q=()=()

p=(0,c) m

Proof of Step 3: For any ∈ ℝ, let ℓ

∈ℝ.

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Then ℓ is the orbit of through ∈ℝ, and (ℓ

) is the orbit of through cos sin∈ ⊂ ℝ.

ⅰ (ℓ) is a closed set if is rational:

∵ Let for some ∈ℤ, where ≠

Let and Then and are points in the line ℓ

, and : In fact, mod and mod Hence (ℓ

) is periodic. In fact, the orbit of starting of arrive at after turning times horizontally, and terning times vertically.

ⅱ (ℓ) is dense in if is irrational.

∵ Let ∈ℝ be fixed, and put ∈ ℝ () (). Then it is enough to show that is dense in ℝ. In fact, if is dense in ℝ, then

∈ℓ

is dense in . By Homework, () = (

∈) is dense in .

Now we shoe that is dense in ℝ. To show this, let ∈ℤ. Then ∈ of and only if ∈: ∵ ∈ ⇔ () () ⇔ for any mod and mod ⇔ for some ∈ℤ and for some ∈ℤ ⇔

⇔ ∈ Since is a subgroup of ℝ, is either discrete or dense in ℝ. Note that is not discrete. In fact, For any ⋯, choose ∈ ℤ such that

∈ . Let . Then is a sequence in ∩ .

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Since is compact and all ′ are distinct, has a limit point, say . Then we have that for any neighborhood of ,

∩≠∅. Suppose is discrete. Then every one point set in is open. Hence has no limit point. The contradiction show that is dense in ℝ. Since , is dense in ℝ.

ⅲ For any ∈ , = the periodic orbit if is rational if is irrational since ⊂ ∩.

2.9. DefinitionThe vector field above is called the rational flow (or irrational flow) on if is rational (or irrational), respectively.

2.10. RemarkAs we have already seen, the -limit of an orbit of an irrational flow on the torus is the whole torus, There are more complex examples of vector field on with rather complicated -limit sets. Meanwhile for the sphere the situation is much simples because of the following topological facts: every continuous closed curve without seq intersections separates into two regions that are homeomorphic to discs (the Jordan Curve Theorem). The structure of an -limit set of a vector field on is described by the Poincaré-Bendixon Theorem.

2.11. Theorem (Poincaré-Bendixon Theorem)Let be a -vector field on the unit sphere with a finite number of singularities. Then for any ∈ , we have ⑴ is a singularity; or ⑵ is a closed orbits; or ⑶ consists of singularities ⋯ and regular orbits such that if ⊂ , then and for some ≤ ≤

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2.12. RemarksThe Poincaré-Bendixon Theorem is not vector field without the hypothesis of a finite number of singularities. For example, let be a ∞ -vector field on given by following figure.

The north and south poles are singularities and the equator is a closed orbit. Let →ℝ be a ∞-function such that i f ∈

i f ∉ ,

Consider the vector field ․, i.e., ․ for any ∈ .Then is a ∞-vector field such that ⑴ every point in is a singularities of ; ⑵ and are singularities of ; ⑶ for any ∈ ∪, consists of infinitely many singularities.

2.13. -topology on ∈ ⑴ is a vector space under suitable addition and scalar multiplication ⑵ -norm on : for any ∈ and , ∥∥ ⇔ and are close ⇔ ⅰ and are -close, i.e., and are -close for all ∈

ⅱ and are -close, i.e., and are -close for all ∈

⋮

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ⅲ and are -close; i.e., and are -close for all ∈

A -topology on is the topology induced by a -norm ∥ ․∥. ⑶ is complete, and so a Baire space;

We say that a topological space is a Baire space if every residual subset of is dense in . A subset of is said to be residual if contains a countable intersection of open and dense subsets of . Note that every complete metric space is Baire, and every complete Hausdorff space is Baire.

⑷ is separable. ⑸ ∞ is dense in for all ≥. ⑹ For any two submanifolds of , we say that and are transversal (⫚) if either ⅰ ∩ ∅, or ⅱ ⊕ for all ∈∩.

The quantitative theory of a dynamical system ∈ consists of a geometric description of its orbit space. Thus it is natural to ask when do two orbit spaces have the same description, the same quantitative features; this means establishing an equivalence relation between dynamical system ∈ .

2.14. DefinitionTwo vector fields ∈ are topologically equivalent ∼ if there exists a homeomorphism → takes orbits of to orbits of preserving their orientation, this last condition means that there is a continuous map ×ℝ→ℝ such that

⑴ for any ∈, the restriction ×ℝ ℝ→ℝ is strictly increasing ⑵ for any ∈×ℝ, .

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We say that is called a topological equivalence between and . Note that "~" is an equivalence relation on .

2.15. HomeworkLet ∈ be topologically equivalent, and let be a topological equivalence between and . For any ∈, ⑴ is singularity of iff is a singularity of ; ⑵ is closed iff is closed; ⑶ ,and .

2.16. DefinitionWe say that a vector field ∈ is -structually stable if there is a

-neighborhood of in such that every ∈ is topologically equivalent to . We say that is structually stable if it is -structually stable.

2.17. RemarksNote that no vector field with a singularity or a closed orbit would be structurally stable if we require the equivalence to be differentiable.

Chapter3. Hyperbolicity

Why hyperbolic?

For any ∈ , we want to study the (topological) behavior of the orbits ∈ℝ of . To do this, we will use the dynamics of the derivate map ; that is, the behavior of the orbit ∈ℝ of . The meet useful property among the dynamics of is the hyperbolicity. For the general definition of hyperbolicity, we introduce following definition (Definition 3.1) and (Theorem 3.2).

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3.1. DefinitionA singularity ∈ of ∈ is said to be hyperbolic if the derivative map → has no eigenvalue of norm 1.

3.2. TheoremA singularity of is hyperbolic if and only if there are constants , and a -invariant splitting

⊕ such that

(1) ∥∥∥∥ for any ∈ and all , and (2) ∥∥∥∥ for any ∈ and all

The above theorem motivates the following definition.

3.3. DefinitionWe say that an invariant set ⊂ is hyperbolic if there are constants , and two subbundles , of such that (1) ∀∈, ⊕⟨⟩⊕ , where ⟨⟩ is the subspace of generated by the vector ; (2) ∀∈, and are -invariant; , =

and = for all ∈ℝ

(3) and are vary continuously with ∈; (4) is exponentially contractive on in the positive direction; , ∀∈, ∀∈, ∥∥∥∥ for all ≥; is exponentially contractive on in the negative direction; , ∀∈, ∀∈, ∥ ∥∥∥ for all ≥; ⇔ max ∥∥, ∥ ∥ ≤ for all ≥. (bundle notation)

We say that is Anosov if is hyperbolic for . In the above definition, we say that : hyperbolic constant of : a skewness of : the stable space at : the unstable space at .

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3.4. Remarks (1) Note that is contractive on in the negative direction if and only if is expansive on in the positive direction (why?)

(2) Show that for each ∈, ∈ ∥ ∥→ →∞, and ∈ ∥ ∥→ →∞.

(3) Geometrically, the hyperbolicity of ⊂ is a structure on the tangent bundle

∈

given by the following figure.

⊕

⊕⟨⟩⊕

(4) Suppose a hyperbolic set contains a singularity and a regular orbit . Then for any ∈, we have

⊕⟨⟩⊕ ⊕ ⊕⟨⟩⊕ Hence we get dim dim ≠ dim dim Consequently, the hyperbolic structure near singularities and near regular orbits are qualitatively different.

(5) If is a hyperbolic set for and is a single chain component for , then either is a single singularity or does not contain any singularities. (because of the assumption that the splitting raises continuously)

3.5. Basic Properties of hyperbolic splitting (1) Extension to the closure: If is hyperbolic for , then is hyperbolic for .

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(2) Robustness: If is hyperbolic for , then there are a neighborhood of and a -neighborhood of such that for any ∈, (i)

∈ℝ is hyperbolic for ;

(ii) and ∈ℝ

are topologically equivalent.

(3) Transversality: If is hyperbolic for , then there exists such that

inf ∡ ∈≥.

(4) Continuation: Let be a hyperbolic periodic orbit of . Then there are a neighborhood U of and a -neighborhood of such that for any ∈,

∈ℝ consists of a single periodic orbit (we will

denote it by ) of Y contained in such that (i) is hyperbolic for ; (ii) index = index ,

where index , denotes the dimension of the stable space . Here, we say that is the continuation of .

3.6. DefinitionLet ∈ and ∈. (1) The stable set of at in given by ∈ → →∞, and the unstable set of at is given by ∈ → →∞.

(2) For any , the -local stable set of at p is given by ∈ ≤ for ≥,and the -local unstable set of X at is given by ∈ ≤ for ≤.

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3.7. Stable Manifold Theorem for Hyperbolic SingularityLet ∈ be a hyperbolic singularity for . Then

(1) is a -immersed submanifold of such that (i) is tangent to at ; (ii) dim = dim. Hence we have

.

(2) There is such that (i)

⊂; (ii)

is a -embedded submanifold of satisfying

.

(3) ∪ ≥ .

Proof. See Reference [2, 3, 4].

3.8. ExampleConsider a flow on given by the following figure

S o u t h e r n

Then is an -immersed submanifold of , but it is not an -embedded submanifold of . Furthermore, there exists such that

is an -embedded submanifold of and

≥

.

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3.9. Stable Manifold Theorem for Hyperbolic SetsLet ⊂ be a hyperbolic set for . Then there exist , , such that for any ∈, (1) is an -immersed submanifold of and

is an -embedded submanifold of such that

; (2) for any ∈

, ≤ ≥

where ℝ→ℝ is a strictly increase function.

Proof. See Reference 5. Invariant manifolds are perhaps the most fundamental objects in modern

theory of dynamical systems, providing the key to study and understand the geometric structure of given systems.

There are several notions extending (uniform) hyperbolicity. Here we introduce two of them : dominated splitting and partial hyperbolicity.

Chap 4. Chain Components and Homoclinic Classes

4.1. DefinitionWe say that a flow is expansive if for any , there is s.tif for any ∈, there is a strictly increasing function ℝ →ℝ s.t (i) (ii) for all ∈ℝ , then ∈ .

Here, is called an expansive constant of corresponding to , and is called a reparametrization. Put ℝ = the set of all reparametrizations.

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4.2. HomeworkIf is expansive, then every singularity of (or fixed point of ) is isolated.

4.3. Definition (pseudo-orbits) (1) TypeⅠ : For any and, , a -pseudo orbit of is a sequence in ×ℝ such that (i) ≥ for all ; (ii) for all .

※ Pseudo-orbits are parts of real orbits.

(2) TypeⅡ : For any and, , a -pseudo orbit of is a mapping ℝ→ such that for any ∈ℝ, for

•

••

※ pseudo-orbits are parts of real orbits.

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4.4. Definition (1) TypeⅠ: We say that a -pseudo orbit of is -shadowed by a point ∈ (or an orbit ) if

there is a reparametrization ∈ℝ such that for all ∈ ,

where

⋯

i f i f ⋯ i f

∈ ⇒ are -close.

(2) TypeⅡ: We say that a -pseudo orbit of is -shadowed by a point ∈ (or an orbit ) if there is a reparametrization ∈ℝ such that for all ∈ℝ.

•• ••

4.5. Definition (1) TypeⅠ: We say that a flow has the shadowing property (or POTP)

on a closed invariant set if for any , there is such that every -pseudo-orbit in is -shadowed by a point in .

(2) TypeⅡ: We say that a flow has the shadowing property on a closed invariant set if for any , there is such that every

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-pseudo-orbit in is -shadowed by a point in .Sometimes, we say that is shadowable for if has the shadowing property on .

4.6. RemarksNote that a compact invariant set is shadowable for under the definition of TypeⅠ if and only if is shadowable for under the definition of TypeⅡ. (See Pilyugin’s book)

4.7. DefinitionA finite -pseudo orbit of is said to be a -chain from ∈ to if and .

* We say that ∈ are chain related ∼ if for any , there are -chain of from to , and from to .

* The set ∈ ∼ is called the chain recurrent set of .

* Clearly, ″∼‶ is an equivalence relation on , and the equivalence classes are called the chain components of .

4.8. HomeworkProve that a subset of is a chain component of if and only if is a connected component of .

4.9. DefinitionWe say that ∈ is a nonwandering point of if for any neighborhood of , and for any , there is such that ∩≠∅.

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Put Ω = the set of all nonwandering points of , = the set of all periodic points of which are not singular, = the set of singularities of X.

4.10. HomeworkProve that

(1) ∪ ⊂Ω ⊂ .(2) Ω are invariant.(3) Ω are closed in , and give an example that is not closed.

4.11. DefinitionLet be the set of all hyperbolic orbits of . We say that ∈ are homoclinically related ∼ if

⫚≠∅ and ⫚≠∅.

•

•

Put ′∈ ′∼′∼

′ .

4.12. RemarksNote that ″∼‶ is an equivalence relation on as we can see in the previous lecture notes (-lemma).

4.13. DefinitionWe say that ∈ is a homoclinic point of ∈ if ∈⫚.

4.14. Theorem = ⫚

= the closure of the set of homoclinic points of = is called the homoclinic class of associated to .

- 34 -

4.15. Basic properties of homoclinic classes of chain components.For any ∈, ⑴ is closed and invariant. ⑵ If ∼ ′ , then ′ . ⑶ Periodic orbits are dense in . ⑷ is a Hausdorff limit of hyperbolic periodic orbits. ⑸ is transitive. ⑹ If is hyperbolic, then it is locally maximal; i.e.,

there is a neighborhood of such that ∈ℝ

. ⑺ If satisfies Axiom A (∪ is hyperbolic), then the nonwandering set can be decomposed into finitely many

basic sets ⊍⋯⊍ such that each is transitive,… Note that every basic set is a homoclinic class. ⑻ Let be the chain component of containing . Then . ⑼ If a chain component is hyperbolic, then there is a periodic orbit in such that . ⑽ If is hyperbolic, then ∪, every basic set is a chain component(=Homoclinic class). ⑾ If consists of a finite number of chain components, then every chain component is locally maximal.

Chapter 5. Vector Fields on Manifolds

5.1. Exponential maps (Boothby, Introduction to Riemannian Manifolds)

5.1.1. LemmaFor any point ∈ and any vector ∈,there is a unique curve → such that and ′.

- 35 -

′

The proof come from the theorem of existence and uniqueness for ODE.

5.1.2. RemarksThe length of the curve from to is

∥′∥

∥′∥

∥∥ ∥∥ . ↑ ·Since is a geodesic, ′ is constant for all ∈ . ∴∥′∥∥′∥∥∥. ·Since has a Riemannian structure, the map ∥′ ∥ →ℝ is continuous. (why?)

5.1.3. TheoremIf is a compact ∞-manifold, then there exists a constant such that ⑴ For any point ∈, the map exp → given by exp is well-defined, where ∈ ∥∥ ⑵ For any point ∈, there is a neighborhood of in such that exp → is a diffeomorphism.

- 36 -

Here exp is called the "exponential map" at ∈.

5.1.4. DefinitionWe say that a geodesic → is minimal if the length of , where .

5.1.5. TheoremIf is a connected, complete Riemannian manifold, then ⑴ For any two points ∈, there exists a minimal geodesic from to .

⑵ Define × →ℝ by = the length of the minimal geodesic from to . Then is a metric on . ⑶ The topology on induced by the metric is equivalent to the original (manifold) topology on .For the proof, see the Boothby's book.

5.2. Orthogonal bundle

5.2.1. NotationLet ∈ be a -vector field on , and let ∈ ≠.For any ∈ , ⑴ We let be the orthogonal (normal) spae to ; that is, ∈ ⟨⟩⟨⟩⊥

: normal space (normal section)

⑵ For any , let ∈ ∥∥ , and let exp⊂ : normal set the normal set at with radius .

- 37 -

5.2.2. TheoremFor any ∈ and any , there are a constant and a -map →ℝ such that ⑴ ⑵ ∈

for any ∈ .

Proof. Homework (apply the Implicit Function Theorem) is called a time function on .

※ Implicit Function Theorem

Q: Consider an implicit function , where ×ℝ→ℝ is

can it be expressed as an explicit function ?

A: Locally “yes” under some condition: More precisely, if ≠ , then there are a nbd of in ℝ, and a -map →ℝ such that

(i) for all ∈

(ii) ′

For example, consider an implicit function :In general, it cannot be expressed as an explicit function :But if ≠ (i.e ≠), then o.k.

- 38 -

→ℝ

↦

⇓ generally <Implicit Function Theorem>Let ℝ× ℝ→ℝ be , and for ∈ℝ×ℝ, ∈ℝ.If ℝ→ℝ is isomorphism, then there is a nbd of in ℝ and a -map → such that

(i) ∀∈

(ii)

∘

= ×-matrix

× ×What is ?Let ℝ× ℝ →ℝ, and let ,,…,:ℝ×ℝ→ℝ be the coordinate function of .

Then

⋯

⋮ ⋮

⋯

ℝ→ℝ is linear.

; Jacobian matrixHow can we define →ℝ and ?For ∈ and >0, let .For simplicity, we identity ℝ: locally.Note that for any , and any ∈,

∈⇔⟨ ⟩ (※)

ℝ

⇓ ℝ

- 39 -

Define × ℝ →ℝ ↦ ⟨⟩⇒ ⅰ) is . ⅱ) ⟨ ⟩⟨⟩. ⅲ) ⟨ ⟩

= ⟨ ⟩ = ⟨ ⟩ ∥∥≠.

By the Implicit Function Theorem, ∃ a nbd of in , say for some , and a -map →ℝ such that

⟨⟩ for all ∈.∴ by (※), ∈, ∀∈.

5.2.3. Definition For any ∈, , the map → is called the poincare map at ∈ and .

각 점 ∈× ℝ에 대하여, Poincare map 이 정의되는 -ball 을 선택하고 에 의존하는 time function 를 다음과 같이 선택할 수 있다.

5.2.4. Lemma Let be fixed. Then there is a continuous function → such that (1) ∀∈ , → is well-defined, (2) ∀∈,

.

- 40 -

5.2.5. Flow box chart∀∈ ∀ ∀ let ≤ ≤ ∈ ⊂ .Define →

↦exp

∥∥

-time length flow box

Theorem For any ∈ and , there is such that the map → ↦ exp is an imbedding.Here, is called the -time length flow box.

5.2.6. Linear Poincare FlowLet

∈

: normal bundle based on .Then we can define a flow on as follows:

×ℝ → ↦

where → is the projection along the flow line ⟨⟩.

•

•

Note : → is given by ∘

where → is the derivative map of .

- 41 -

⟹ ×ℝ→ is a continuous flow on . (i) is continuous (ii) (iii)

= the linear poincare flow on induced by ∈ .

Q : What is the relationship between the poincare map and the linear poincare flow?

A : The derivative map of poincare map = the time -map of the linear poincare flow : that is,

∀∈∀∈ℝ,

= . (Homework: use the chain rule)

Note : ⊂ → ⊂ , -into diffeo. ≤→ ≤ , linear,

where .

∞manifold of dimension dimension subspace of .

•

•

: linear

: -diffeo

HomeworkLet → be a diffeomorphism(into), and let → be linear. Then .「∵ ∀∈ ,

- 42 -

∴ by chain rule

t-function

․

․

․

∴ ․

∴ ∀∈

․

․ real number

→ℝ

⋯

⋮

상수

∴ 」

•

•

․

∴ is the projection of along the flow line ⟨⟩.∴

.

5.3. Hyperbolicity and Dominated splitting for linear Poincar flowsThe theory of the hyperbolic dynamics has been extremely successful. Hyperbolicity characterize the structurally stable systems ; it provides the structure underlying the presence of homoclinic orbits ; a large category of rich dynamics are hyperbolic(geodesic flow in negative curvature, linear automorphisms, mechanical systems, etc).The hyperbolic theory has been fruitful in developing a geometrical approach to dynamical systems, and under the assumption of hyperbolicity we obtains a

- 43 -

satisfactory (complete) description of the dynamics of the system from a topological and statistical point of view.

Let ∈ , ⊂ an -invariant set, and denotes the linear poincare flow on

∈

induced by .

5.3.1. Definition An -invariant splitting of ,

⊕ is called a hyperbolic spiltting

if there are constants and such that for any ∈, ∥ ∥≤ and ∥ ∥≤

for all ≥.

Geometric meaning ; Normal bundle over can be splitting into two parts ; a contracting part

and an expanding part .

There are relaxed forms of hyperbolicity such as partial hyperbolicity, dominated splitting, etc.

How to relax?

⊕ is a hyperbolic splitting.

⇔ For any ∈, any non-zero vectors ∈ and ∈, ∥ ∥→ as →∞, and

∥ ∥→∞ as →∞. →↚ ∥ ∥

∥ ∥→ as →∞ (※)

Now, let us consider another splitting ⊕ which satisfies the property (※) for ∈ and ∈, ∈ ;So, we introduce a relaxed form of hyperbolicity called a “dominated splitting” as follows.

5.3.2. DefinitionAn invariant splitting of , ⊕, is called a dominated splitting if there are constants and such that

- 44 -

for any ∈, any unit vectors ∈ ∈,

∥ ∥∥ ∥

≤ for all ≥.In the case, we say that is dominated by , and denoted by ≺.

5.3.3. HomeworkLet ⊕ be a invariant splitting. Then the following conditions are pairwise equivalent. ⑴ ⊕ is a dominated splitting,

∈

∈

. ⑵ There is such that for any ∈, any unit vectors ∈ ∈ ∥ ∥

∥ ∥≤

if ≥ . In this case, we say that ⊕ is a dominated spliiting. ⑶ There are constants and such that for any ∈, ∥∥․∥ ∥≤

⑷ There is such that for any ∈, ∥∥․∥ ∥≤ if ≥ . ⑸ There are constants and such that for any ∈,

∥∥≤ for all ≥ ,

where inf ∥∥ ∥∥ is the minimum norm of . ⑹ There is such that for any ∈,

∥∥≤

if ≥ .

•

•

5.3.2. Definition⊂ : - invariant set ⊕ is a dominated splitting for ⇔ ∃ and such that

- 45 -

∀∊, ∀unit vectors ∊, ∊,

∥∥∥∥

≤ for all ≥.

The followings are pairwise equivalent. ⑴ ⊕ is a dominated splitting for . ⑵ ∃∊ℕ s.t. ∀∊, ∀unit vectors ∊, ∊, ∥∥

∥∥≤

if ≥ . ⑶ ∃ and s.t. ∀∊, ∥ ∥․∥ ∥≤ for all ≥. ⑷ ∃∊ℕ s.t. ∀∊, ∥ ∥․∥ ∥≤ if ≥ . ⑸ ∃ and s.t. ∀∊,

∥ ∥≤ for all ≥

⑹ ∃ s.t. ∀∊,

∥ ∥≤

if ≥ .

Homework⑴ : ⊕ is a dominated splitting for ⇔ ∃ and s.t. ∀∊, ∀unit vectors ∊, ∊,

∥∥∥∥

≤ for all ≥

⇔ ∃ and s.t. ∀∊, ∀non zero vectors ∊, ∊,

∥∥∥∥

≤∥∥∥∥ for all ≥

⑶ : ∃ , s.t. ∀∊, ∥ ∥․∥ ∥≤ for all ≥, where .

- 46 -

⑴⇒⑶ : ∀∊, ∀≥, ∀unit vectors ∊, ∊, Let .

Then and ∥∥. By ⑴, ∥∥

∥∥≤∥∥

∥∥∥∥

․

∴ ≤ ․∥∥

⇒ ∥∥․∥∥≤∥∥․∥ ∥≤

Since ∊, ∊ are arbitrary, ∥ ∥․∥ ∥≤ for all ≥. This proves ⑶.

⑷⇒⑶ : ∀≥, choose ∊ℕ∪ such that ≤ . ∥ ∥․∥ ∥≤∥ ∥․∥ ∥․∥ ∥․∥ ∥ ≤ ∊

∥ ∥∥ ∥ ․

≤ ․

․ ,

,

The following theorem which is crucial to get the hyperbolicity of an -invariant set for vector fields was proved by Doering(1987).

- 47 -

5.3.3. TheoremLet ⊂ be a closed -invariant set. Then is hyperbolic for if and only if the LPF restricted to has a hyperbolic splitting

⊕ .

To prove the theorem, we use the notion of the angle between two linear spaces.

5.3.4. Remarks ⑴ If is hyperbolic for , then there is such that

∡ ∊≥ , ∡ ⟨ ⟩ ∊≥ , and ∡⟨ ⟩ ∊≥ .

⊕⟨ ⟩⊕

⟨ ⟩

⟨ ⟩

the angles among subspaces , , ⟨ ⟩, ∊, are uniformly bounded from zero. In this case, is called a transversal constant of .

⑵ if admits a dominated splitting ⊕, then ∡ ∊≥ . Let , be subspaces of such that ⊕. What is the angle ∡ ?

∡ def=

∡ ∊ ∊

- 48 -

∴∀∊ ∊

∡ ≥.There are many ways to characterize the angle ∡ .

Proof of Theorem 5.3.3.⇒) Suppose is hyperbolic for . Then for any ∊, there is a hyperbolic splitting

⊕⟨ ⟩⊕ such that is contracting on and is expanding on . That is, there are and such that ∥ ∥≤ and ∥ ∥≤. Let

→ be the projection along the flow has ⟨⟩, and put

⟨⟩⊕⊕ ⟨⟩⊕ ⟨⟩⊕⊕

Consider the following diagram:

∆ ∆

∆

∘∘

Then the Diagram is commute. 『∵ Let . Note: and ⟨⟩ are -invariant. Since , is a subspace of ⊕⟨⟩.

- 49 -

•

∈

∆

•

∆

∀∈ , let . ⇒ ∃∈⟨⟩ (∵ is a subspace of ⊕⟨⟩) ⇒

⇒ ∴

∴ ∘ ∘ on . 』

Note that → ≤ is linear and ∥∥≤. ()

•

『∵ Since ⊕⟨⟩, ∀∈ ∃′∈ s.t. ′ Then ′ ∴∥∥∥′∥≤∥∥ ∴∥∥sup∥∥≤

∀∈, ∃′′∈ s.t. ′ ′ ∴ ′′ ′′

→ is a linear, when : vector space ⇒ For any subspace of ,

- 50 -

is a subspace of . ∴ → is linear, are subspace of . ⇒ are subspace of . Moreover, ⊕

⊕

⊕ . 』

Then →

is also linear. ⇒∥ ∥∞ ?

『∵Let be a transversal constant of . Then the angle ∡ ⟨⟩≥ , ∀∈. ∈

∴∀∈ with ≠ ∥∥∥∥

≥ sin. ∴∥∥∥∥

≤sin .

∴∀∈ with ≠, ∃∈

s.t. , ⇒

∥ ∥≤sin

.

∴ ∥ ∥sup∥∥∥ ∥

∈≤sin

, ∀∈.』 ∴ ∥ ∥≤ ∥∥∘∥∥∘∥ ∥

≤ ․ ․sin

≤ for all

when ․sin .

⇐ Suppose is hyperbolic for . Then there is a -invariant splitting

⊕ such that

there are and satisfying ∥ ∥≤ and ∥ ∥≤ for any ∈ and all ≥.

- 51 -

Let ∥∥ ∈∥∥ ∈ .

Then ∞ since ∥․∥: →∞↦∥∥ is continuous.

『∵ has a Riemannian structure and so ∥․∥ vary continuous on ∈.』 Choose such that , and let

. Since

is -invariant,

⊕⟨⟩ ∪∈

⊕⟨⟩ is -invariant.

•

•

∆ ∆

∀∈⟨⟩, . Then

∈⊕⟨⟩. ∴

⊕⟨⟩ is -invariant. First, for each ∈, we will find a subspace. of ⊕⟨⟩ which is -invariant; i.e.,

Since ⟨⟩ is -invariant, for each ∈ and , the linear map ⊕⟨⟩

⊕⟨⟩→⊕⟨⟩can be expressed as a matrix of the following form;

⊕⟨⟩

, ∈, ∵ Choose a basis ⋯ of ⊕⟨⟩ and

a basis ′ ′′⋯′ ′ of ⊕⟨⟩. Let

be the matrix of w.r.t. and ′.

Since ⟨⟩ is -invariant, . In fact, for any nonzero vector ∈⟨⟩,

- 52 -

∈

⊕⟨⟩

∴ for any ∈⟨⟩. ∴ .

Note that ⑴

. ∴∥∥≤

∵ ∀∈,

since ∈⟨⟩

⑵ ∥∥∥ ∥ ∈≤ ∥∥ ∈∥∥ ∈ .

∵ ∀∈⟨⟩,

⟨⟩ ⟨⟩→⟨⟩ is an isomorphism. ∴∥∥∥⟨⟩∥∥∥∥∥

≤∥∥ ∈∥∥ ∈

•

•

∥ ∥∥⟨⟩∥∥∥∥∥≤∥∥ ∈

∥∥ ∈.

Now we will find a linear bundle map →⟨⟩ such that

ⅰ for each ∈, →⟨⟩ is linear; ⅱ = the graph of , where :

- 53 -

•

•

•

∆ ∆ ∆

∆ ⊕

: graph of is -invariant. : graph of is -invariant.

⋮

↦

a graph transform.

How can we find →⟨⟩ such that the graph of is

-invariant for each ∈? Apply the generalized contraction mapping theorem! Let = the graph of →⟨⟩ Then is a subset of ⊕⟨⟩.

Note that for any ∈, is -invariant ⇔∀∈ for ∈,

⋯ ※

since ※ belong to

⇔ ∀∈

⇔ ∘ ∘

⇔ ∘∘

∴ ∀∈, ∘∘ 를 만족하는 linear map →⟨⟩

를 찾자:⇔ Bundle Notation을 쓰면 ∘∘를 만족하는 lenear bundle map

- 54 -

→⟨⟩를 찾자:

이런 lenear bundle map 를 찾기 위하여 generalized contractive mapping theorem을 이용할 것이다.

5.4. Generalized Continuous mapping theorem

Homework (Generalized Contraction Mapping Theorem)Let be a complete metric space, and let ×→ be a uniform contraction on the second factor; for any ∈, the map → defined by is a contraction.Then there is a unique map → such that for all ∈.Moreover, we have that

(i) if is continuous, then is continuous(ii) if is Lipschitz, then is Lipschiz(iii) if is , then is

(In this case, we assume that : Banach spaces)

To apply the contraction mapping theorem, we let

⟨⟩ = ∈ ⟨⟩ for ∈ = the set of all linear bundle maps from

to ⟨⟩.

⟨⟩ = ∥∥≤, where ∥∥ sup ∥∥ ∈⟹

⟨⟩∥․∥ is a complete metric space, and

⟨⟩ is a closed subset of ⟨⟩.

Define ⟨⟩ →

⟨⟩ ↦ ∘ ∘

⟹ (i) is well-defined: Put sup ∥∥ ∈, 1, and

Then for any ∈ ⟨⟩,

∥ ∥∥ ∘∘∘∥ ≤∥ ∥∥∥∥∥∥ ∥∥∥ ≤ ∥∥ =∥∥

- 55 -

≤

(※∥∥≤ )

∴ ∈ ⟨⟩.

(ii) For each ∈, the map →⟨ ⟩ is a contradiction For any , ∈

⟨⟩, ∥∥ ∥ ∘∘∘∘∥ ∥ ∘∘∥ ≤ ∥ ∥∥∥∥∥ ≤∥∥ ∥∥∥∥ ∴ By the generalized contraction mapping theorem, there is a unique ∈

⟨⟩ such that ∘.

For each ∈, let be the graph of ⟹ (i) is -invariant for all ≥. By our contraction, ∀∈, . Then

; is -invariant. 」 Moreover, for any ≥,

.

∴ is -invariant ;

Also is -invariant by our contraction.

∴

Again, by the uniqueness of , ⇔ the graph of is -invariant ⇔ is -invariant ⇔ is -invariant for all ≥.

- 56 -

(ii) For each ∈, is contracting on . ∀∈

with ∈,

∴ , Since ∥∥≤ , ∥∥→ as →∞.

(iii) More precisely, there are constant , ′ such that ∥∥′ for all ≥. First, we have : ∀∈ , ∥∥∥ ∘ ∥≤∥∥∥∥

≤ ∥∥ This implies that ∥∥≤

∥∥ ⇒∥∥≤ Choose such that

.

Then ∥ ∥≤

Let sup∥∥ ≤ ≤, and let .

Then and . For any ≥, choose ∈⋯ and ≤≤ such that

.

•

•

•

•

Then ∥∥≤ ∥∥․∥∥ ≤

․∥∥ ≤

․

.In the same way, We can construct such that ․ dim dim

, ∈. ․∥ ∥≤

, ≥

- 57 -

Finally ⊕⊕⟨⟩ for all ∈. ․ dim dim dim

dim dim.

․ ∩ ∀∈. Define

→ is an isomorphism

↦

< Contradiction mapping theorem>

Definition Let be a metric space.We say that → is a contraction if there is a constant such that ′≤ ′ for any ′∈.Here, is called a Lipschitz constant of .

Theorem 1 (Contraction mapping theorem) Let be a complete metric space.If → is a contraction, then has a unique fixed point.

DefinitionLet be a set, and let be a metric space.We say that ×→ is a uniform contraction on the second factor if there is a constant such that for each ∈, the map → defined by is a contraction with Lipschitz constant .

Theorem 2 (generalized contraction mapping theorem).Let be a set, and let be a complete metric space.If ×→ is a uniform contraction on the second factor, then there is a map → such that for all ∈.Here is called the fixed point map of .Proof: For each ∈,

→ has a unique fixed point, which we denote by .This defines the fixed point map → of satisfying for all ∈.Now we investigate the extent to which properties of influnce properties of :

- 58 -

Which properties of properties of : Banach spaces (normed vector space which is complete)

(1) If ×→ is continuous, then the fixed point map is continuous.

(2) If is Lipschitz, then is Lipschitz then is . (3) If is , then is .

Proof of (1),(2),Since F is a uniform contraction on the 2nd factor, there is a constant such that for any ∈,

′≤′ for all ′∈.For any ′∈, ′ ′ ≤′ ′′′ ≤′ ′∴ ′ ≤

′ → as →′ (∴ is conti)

(1) If →′, then →′.(2) Since is Lipschitz ; i.e.,∀′ ′≤′, ∃ such that ′≤′ : ∴′≤

′ .

To prove (3), we need the notion of -derivative :Let be Banach space (normed vector spaces which are complete), and → is differentiable at ∈ ⇔ ∃a linear map → such that lim

→∥∥∥∥

(※).Note that if L exist, it is unique (why?)We call it, the derivative of at ∈; and denoted by ;If exists for all ∈, We say that is differentiable.Let be the set of linear maps from to with a norm ∥․∥defined by ∥∥ sup ∥∥∥∥ sup∥∥

∥∥∥∥≠.

Then ∥․∥ becomes a Banach space.If is differentiable, then the map

→ is called the derivative of . ↦

- 59 -

Note ․ If is Lipchitz with constant , and is differentiable at ∈, then ∥∥≤ (why?). ․ ×→ is differentiable at ⇔ lim

→∥∥∥ ∥ (※)

․ Higher derivatives are defined inductively by , ≥

․ We say that is if is continuous, is ∞ if is for all ≥

․ →

↦ → ↦ →

․ the space of linear maps from to with sup norm ∥․∥ by ∀∈,

∥∥sup∥∥∥∥, where ∥∥sup∥∥∥∥. the space of bilinear maps ×→ with sup norm ∥․∥

by ∥∥sup∥∥∥∥∥∥. Then ≃ : isomorphic by ↦ s.t ↦

In general, we have ≃ .

More generally, ×→ is ⇒ ∀∈×, (why?)

Proof of (3)Since ×→ is lipschitz on the 2nd component with constant and is differentiable, ∥∥≤ for any ∈× .Hence → is an isomorphism (why?)

『∵ Lipschitz()≤ ⇒ → is injective linear ∵ ker ; ⇔∴≠. 』First we show that is differentiable with derivative ․, ∀∈.

- 60 -

Claim: lim →∥∥∥ ∥

∥ ∥ ∥ ․∥ ∥ ∥ ≤ ∥∥∥∥ ≤ ∥∥∥ ∥

(※) note ∥∥≤ ∥∥ since is locally Lipschitz (∵ is locally Lipschitz )

× is differentiable at ⇔∥∥

∥∥→ as ∥∥→

Since is differentiable at , ⇔∥∥

∥∥→

as∥∥→

If ∥∥→ , then ∥∥≤∥∥→

∴ ∥∥→ . ∴ is differentiable at ∈.

Next can be expressed as a composite:

- 61 -

× ×

××

×

×

∘

where the ball w ith center 0 and radius in , and

→ is ∞. ↦

< Appendix >

Tangent Bundle Dynamics • →, a -vector field, ≥

• ×ℝ→, the flow induced by : i.e., ∀∈×ℝ ,

•

•

• will be denoted by := ∈ℝ : one-parameter group of -diffeomorphisms on = a transformation g = ∈ℝ. • For simplicity, ∈ℝ will be denoted by . • Consider the flow on induced by ∈ℝ := ∈ℝ

×ℝ ↦

⇒ is the flow on induced by vector field ․

- 62 -

․

∘

•

•

•

***

ℝ

• •

∈

∈

≈ ∈

ℝ ≈×ℝ

∈ℝ or ∈ℝ

∈ℝ will be denoted by why? : a vector field on its flow on : a vector field on its flow on

flow on induced by

• why tangent bundle dynamics?

- 63 -

Chap 6. Fundamental Theorems for Flows

1. Stable Manifold Theorem2. Shaodwing Lemma3. Tubular Flow Theorem4. Hartman – Grobman Theorem : Hartman, Grobman5. -Lemma (or Inclination Lemma)6. -closing Lemma : C.pugh7. -connection Lemma : Hayashi, -stability theorems8. Ergodic closing Lemma : Mane, Wen9. Perturbation Lemma (Franks Lemma) : Franks, Bonatti

6.1. Stable Manifold Theorem

6.1.1. Theorem For any hyperbolic set of ∈ , there are constants , and such that for any ∈, (1)

is an -embedded submanifold of , and is an -immersed submanifold of such that

;

(2) for any ∈, there is a strictly increasing function ℝ →ℝ

such that ≤ for all ≥.

6.1.2. Remarks Invariant manifolds

are fundamental objects to

understand the geometric structure of given system.For the proof : use graph transform method : <Irwin’s book> (646p-659p) : diffeo → flow <palis’s book> (75p-80p) : diffeo

- 64 -

6.2. Shadowing Lemma

6.2.1. Theorem Let ⊂ be a hyperbolic set of . then is shadowable for ; i.e., for any , there exists such that every -pseudo orbit in is -shadowed by a point in : • Anosov (1970) • Bowen (1975) : geometric proof • Different proofs of similar statements were given by conley ets..

6.2.2. RemarksIt is a lemma describing the behavior of pseudo-orbits mean a hyperbolic set. It says that every pseudo-orbits stays uniformly close to some true orbit : For proof, refer the Pilyugin book.

6.3. Tubuler Flow Theorem (local linearization thm)It says that the local behavior of orbits near a regular orbit is very simple : it can be linearized.

6.3.1. TheoremLet ∈ be a regular point of , and let ∈ℝ .Let be a vector field on given by ⋯ .Then there is a nbd of in such that

is topologically equivalent to ; i.e., there is a homeomorphism → which takes orbits of on to orbits of on .

•

Linearization of orbits near a regular point

For proof, refer the Palis’s book (p40).

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6.4. Hartman–Grobman TheoremIt says that a vector field is locally equivalent to its linear part at a hyperbolic singularity.

6.4.1. TheoremLet ∈ be a hyperbolic singularity of .Then there are a nbd of and a nbd

of in such that is topologically equivalent to .

• the proof is valid in Bowen’s book • for proof, see Palis’s book (60p–63p)

6.5. Inclination Lemma (or -Lemma)

6.5.1. TheoremLet ∈ be a hyperbolic singularity of .

⋯

•

•

The inclination lemmaFor proof : refer the Palis’s book (p80-p88).

<for diffeo and for flow case>.

Take an embedded disk in which is a nbd of in , and choose a nbd of in .Let be a transverse disk to at s.t dim dim.

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For any ≥, let be the connected component of ∩ which contains .Then for any , there exists such that for any ≥, is -close to in the -topology : : inclusions ⇒ .

An application of -lemma

6.5.2. Theorem Let ∈ be a hyperbolic singularity (or periodic orbit) of .⇒ ⫚ is topologically transitive ; i.e., for any open sets in , there is ≥ such that ∩≠∅.

Proof. Let be nonempty open sets in .Then we have ∩⫚≠∅ and ∩⫚≠∅.Take ∈⫚ and ∈⫚.

in

in

By the -lemma, we can choose a point such that ∈ for large ≥, ∈ for large ≥.

Then ∈.

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∴ ∈ . ∴ ∈ ∩≠∅. ∴ ∩≠∅.

※ Suppose is a saddle connection.

and do not have the transverse intersection : ∴⫚∅ .

Recursive Sets : Let ∈ .

1. Limit set : ∀∈ ∈ → for →∞, and ∈ → for →∞ .

∈

, ∈

. ∪ .

2. Nonwandering set : ∀∈ nonwandering ⇔∀ ∃≠∈ℝ ∩≠∅. the set of all nonwandering point of .

3. Chain-recurrent set :

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Homework : (1) ⊂ ⊂ ⊂ . (2) Given these examples to show that ≠ ≠ ≠ . (3) and are closed and -invariant.

6.6. -closing Lemma.

6.6.1. Theorem ( -closing Lemma, Pugh)∈ : a nonwandering point⇒∀ of , ∀ of ,

∃∈ such that ∩≠∅

application

6.6.2. Theorem -generically, ,

where =the set of periodic points and singularities of .i.e., ∃a residual set ℛ⊂ s.t. ∀∈ ℛ ,

.i.e., ∃a collection of open dense subsets of s.t.

∈ ⊃

∞

: a residual set.Proof. Let

∞ be a countable basis of .For each ∈ℕ, we letℱ ∈ ∩ ∅, andℋ ∈ ∩≠∅.

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⇒ Step 1 : ℋ is open in .「∵ ∈ℋ

⇒∃a hyperbolic orbit of s.t. ∩≠∅

⇒ By the stability of hyperbolic set,∃ a -nbd of and a nbd of s.t.∀∈

∈ is a hyperbolic periodic orbit of ,

which is called the of w.r.t. .Then ∩≠∅ (if necessarily we shrink as small as possible.)⇒ ⊂ℋ . 」

U For each ∈ℕ,

i) ℋ∪ℱ∘ is open in , and

ii) ℋ∪ℱ∘ is dense in .

Step 2 :「∵ ∀∈ , ⇒ ∀ ∩≠∅)

if ∈ℋ∪ℱ ,∃a periodic orbit of s.t.· is not hyperbolic; i.e.,

의 eigenvalue가 1인 것이 존재⇒ Under small perturbation, we may assume 의 eigenvalue≠

·∩≠∅

⇒∀ -nbd of , ∃∈ s.t. is a hyperbolic periodic orbit .

『∵ : non-hyperbolic periodic orbit of

⇒ by the Franks lemma, ∃∻ s.t. is a hyperbolic periodic orbit of .』

⇒∈ℋ .∴∩≠∅ 」

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Step 3 : ℛ

∞

: residual set in .∀∈ℛ, .

「∵ Let ∈

For any nbd of , ∃∈ s.t. ∈ ⊂

Since ∈ ℋ∪ℱ∘ : -nbd of , : nbd of .

By the -closing lemma,∃∈ℋ∪ℱ

∘ s.t. ∩≠∅

⇒∈ℋ 「∵ if ∈ℱ

∘ , ∃⊂∘

⇒∀∈

∩ ∅ but ∩≠∅

∴∈ℋ 」 ⇒ ∩≠∅

∴ ∩≠∅

Since is an arbitrary nbd of ∈ . 」

improved by the -Connecting Lemma.

6.7. -connecting Lemma

6.7.1. Theorem ( -Connecting Lemma, Hayashi, 1997)∀∈ with ∩ ≠∅

∃∻ s.t. ∈

· ·

· Hayashi (1997), Ann, of Math. 145, 81P-137P.· Application : It is used to prove the -stability conjecture for flow posed by smale and Palis.

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6.7.2. Theorem ∈ is structurally stable ⇔ , satisfies i.e.,

․ ∪

․ is hyperbolic satisfies the strong transversality condition(STC); i.e.,

·for any ∈

is transversal to

The following improved -Connecting Lemma was proved by Wen and Xia (2000), Trans. AMS, 5213-5230.

<Improved Connecting Lemma>More precisely,

···

·

··

or or

and have orbits which visit a given small nbd of ∉ and they are for away from a piece of negative orbit of .⇒∃∻ s.t. are in the same orbit of

6.7.3. Theorem (Improved Wen, 2000)Let ∈

For any of there are constants and s.t.for any and any two points ∈ satisfying

(1) ∉

(2) ∩

≠∅

(3) ∩

≠∅ ,there is ∈ such that

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(1) ∈ on

(2) ∈.

6.8. The Ergodic Closing LemmaFor any ∈ , every orbit of can be well-approximated by periodic orbits of -nearby flows

More precisely,

6.8.1. Definition For any a non-singular point ∈ of is said to be strong closed (or well closable) if for any nbd of ,

∃∈ ∈ and such that

on

for all ≤ ≤

· ·

·

Put ∈ ∀ is -strongly closed = the set of all -strongly closed points of for any sufficiently

small

6.8.2. Theorem (Ergodic Closing Lemma, Flow version)For any invariant Borel probability measure on ∪

( is -invariant ⇔ is measure preserving)

Homework is -invariant. is a borel set :

·Complete union of open sets(or closed sets).·Countable intersect of open sets(or closed sets).·Relative complement.

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Main feature of this approximation : It provides a bound for the distance between the original orbit

and the approximating periodic orbit . However it cannot applied to every orbit, just to a total probability

subset. Such amount of control is not allowed to by the -closing

lemma and -connecting lemma.

: a Borel measure on . : a l-cpt space. : the smallest -algebra containing topology of

·∈

·finite intersect ·countable union closed ·complement

Any measure defined on is called a borel measure.

6.9. Perturbation LemmaThe flow version of Frank’s lemma implies that : ∀∈ ∀regular point ∈ , ∀ -nbd of ∀ ∃ s.t.every -perturbation of the linear Poincaré map →

can be expressed as the linear Poincaré map of a -perturbed field ∈

More precisely,

6.9.1. Definition (-pertubation of ) A linear map →

is said to be an -pertubation of the linear Poincaré map if there is a linear map → such that

∥∥

∘

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·

·

→

↦

6.9.2. Theorem (Flow version of Frank’s Lemma, 2006, Bonatti)∀regular point ∈ , ∀ -nbd of ∀ ∃ such that∀tubular nbd of ∀-perturbation of ∃∈ such that

on ∪ the linear Poincaré map of w.r.t. ∈×ℝ, where

the collection of

min ∈

min ∈

∻

Since on

and

-small perturbation along the orbit segment

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The perturbation lemma has lots of applications : For example,

6.9.3 Theorem (Bonatti, 2006, ETDS, 1307-1337)Every periodic orbit of with sufficiently large period

either a admits a dominated splitting, or Can be turned into a sink ou source by a -small perturbation along the orbit.

: sink : source

6.9.4. DefinitionWe say that is robustly transitive if there are a -nbd of and a nbd of s.t.

∈ℝ

: is locally maximal (or isolated), ∀∈

∈ℝ is transitive.

6.9.5. Theorem (Bonatti, 2006, ETDS, 1307-1337)If is robustly transitive for ∈ ,then the linear Poincaré flow of on admits a dominated splitting.

Note that we do not know admits a dominated splitting over .

References1. S. Bautista, C. Morales, Lectures on Sectional Anosov flows, Monographs, 2010.2. C. Bonatti, L. Diaz, and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Springer-Verlag,

2005.3. W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic

Press, 1975.4. M. Hirsch, Differentiable Topology, Springer-Verlag, 1976.5. M. C. Irwin, Smooth Dynamical Systems, Academic Press, New York, 1980.6. J. Palis and D. Melo, Geometric Theory of Dynamical Systems, An Introduction,

Springer-Verlag, 1982.7. S. Pilygin, Spaces of Dynamical Systems, Berlin-Boston, 2012.8. C. Robinson, Dynamical Systems, CRC Press, New York, 1999.9. M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, 1987.10. S. Smale, Differentiable Dynamical Systems, Bull. Amer. Math. Soc. 73 (1967), 747-817.