38.03.03 · dynamics / hyperbolicity-structural-stability

The Hadamard-Perron Stable and Unstable Manifold Theorem

shipped3 tiersLean: none

Anchor (Master): Katok-Hasselblatt 1995 *Introduction to the Modern Theory of Dynamical Systems* (Cambridge) Ch. 6 §6.2 (the Hadamard-Perron theorem, graph transform, $C^r$ section theorem) and §6.4 (the inclination / $\lambda$-lemma); Hirsch-Pugh-Shub 1977 *Invariant Manifolds* (Springer LNM 583); Shub 1987 *Global Stability of Dynamical Systems* (Springer) Ch. 5

Intuition Beginner

Stand at the very top of a smooth mountain pass — the low saddle between two peaks. Roll a ball and watch what happens. Almost every push sends it tumbling down one of the two valleys. But there are exactly two special lines of approach along which a ball, perfectly aimed, rolls toward the pass and slows to a stop right at the top. And there are two special lines along which a ball, given the tiniest nudge, rolls away from the pass forever. The aim-and-arrive lines form the stable manifold; the nudge-and-leave lines form the unstable manifold.

A dynamical rule with a saddle behaves the same way. Near the balance point, some starting positions are funnelled in by repeated application of the rule, their distance to the balance point shrinking to zero. Others are flung out, their distance growing. The set of all in-funnelled points is a smooth curve or surface — the stable manifold — and the flung-out points fill another smooth surface, the unstable manifold. The two cross only at the saddle itself.

Why does this matter? Because these two surfaces are the skeleton of the whole picture. Once you know them, you know which way every nearby trajectory will eventually go. They organise the flow the way ridgelines and valley floors organise water running off a mountain — and the remarkable fact is that they are always smooth, never jagged, no matter how the rule bends space around the saddle.

Visual Beginner

Picture a saddle point at the centre of a small square patch. Through it run two curves crossing like an X. One arm of the X is the stable curve: dots placed on it march inward toward the centre, step by step, getting closer and closer. The other arm is the unstable curve: dots on it march outward, away from the centre. A dot placed off both curves first slides toward the stable arm, then peels away along the unstable arm — tracing a path shaped like the branch of a hyperbola.

A small table fixes the vocabulary.

name	what the rule does to points on it	everyday image
stable manifold	pulls them in toward the saddle	rolling toward the mountain pass
unstable manifold	pushes them out from the saddle	rolling away down a valley
tangent direction	the straight line each curve hugs at the saddle	the aim-line you'd draw first
saddle point	the single crossing point of the two curves	the top of the pass

Worked example Beginner

Take the simplest saddle rule on the plane: a point $(x, y)$ becomes $(x /2, 3 y)$ . The first coordinate is halved every step; the second is tripled. The balance point is the origin $(0, 0)$ , which stays put.

Step 1. Track a point on the $x$ -axis, say $(8, 0)$ . Applying the rule: $(8, 0) \to (4, 0) \to (2, 0) \to (1, 0) \to (0.5, 0)$ . The distance to the origin halves each time, marching to zero. Every point of the $x$ -axis is pulled in, so the $x$ -axis is the stable manifold.

Step 2. Track a point on the $y$ -axis, say $(0, 1)$ . Applying the rule: $(0, 1) \to (0, 3) \to (0, 9) \to (0, 27)$ . The distance triples each step, flying off to infinity. Every point of the $y$ -axis is pushed out, so the $y$ -axis is the unstable manifold.

Step 3. Track a mixed point, $(8, 1)$ . The orbit runs $(8, 1) \to (4, 3) \to (2, 9) \to (1, 27) \to (0.5, 81)$ . The $x$ -part shrinks toward zero while the $y$ -part blows up. The point first drifts toward the $y$ -axis, then races up it.

What this tells us: the in-funnelled points form one straight line and the out-flung points form another, crossing only at the origin. For this linear rule the two manifolds are exactly straight lines. The theorem of this unit says that when you bend the rule into something curved — adding higher-order wiggles — the two manifolds bend with it into smooth curves, but they still cross only at the saddle and still hug those same two straight directions right at the crossing.

Check your understanding Beginner

Exercise (easy, multiple choice).

The stable manifold of a saddle point is best described as which set of points?

A. The points that stay exactly fixed under the rule. B. The points whose orbits march toward the saddle, with distance shrinking to zero. C. The points whose orbits fly away from the saddle. D. The points that return to where they started after one step.

Hint

"Stable" means trajectories settle down — they approach the balance point as the rule is applied again and again.

Answer

B. The stable manifold is the set of starting points whose forward orbits converge to the saddle: their distance to the saddle shrinks to zero. Option C describes the unstable manifold. The saddle itself is the single fixed point (A), but the stable manifold is a whole curve or surface through it, not just that one point. (D) describes period-one points, which for a saddle is just the saddle.

Formal definition Intermediate+

Let $M$ be a smooth Riemannian manifold, $f : M \to M$ a $C^{r}$ diffeomorphism ( $r \geq 1$ ), and $Λ \subseteq M$ a hyperbolic set with splitting $T_{Λ} M = E^{s} \oplus E^{u}$ and uniform constants $(C, λ)$ , $0 < λ < 1$ , as in 38.03.01. The point $x \in Λ$ need not be fixed; write $f^{n}$ for the $n$ -th iterate and $d (\cdot, \cdot)$ for the Riemannian distance.

Definition (local stable and unstable sets). For $ε > 0$ the local stable set and local unstable set of $x$ are $$ W^s_\varepsilon(x) = {, y : d(f^n x, f^n y) \leq \varepsilon \text{ for all } n \geq 0 ,}, \qquad W^u_\varepsilon(x) = {, y : d(f^{-n} x, f^{-n} y) \leq \varepsilon \text{ for all } n \geq 0 ,}. $$ The global stable and unstable manifolds are $$ W^s(x) = {, y : d(f^n x, f^n y) \to 0 ,} = \bigcup_{n \geq 0} f^{-n}\big(W^s_\varepsilon(f^n x)\big), \qquad W^u(x) = \bigcup_{n \geq 0} f^{n}\big(W^u_\varepsilon(f^{-n} x)\big). $$

Sign / convention. Forward time contracts $E^{s}$ and expands $E^{u}$ , matching 38.03.01 and Katok-Hasselblatt ^{[Katok-Hasselblatt 1995 §6.2]}. The local stable set is the set of points whose forward orbit shadows that of $x$ within $ε$ ; the contraction on $E^{s}$ forces such an orbit to converge to that of $x$ . This is the global, uniform-over- $Λ$ form of the local stable subspace of a Hurwitz linearization 02.12.08.

Definition (graph over a subspace). Fix $x$ and identify a neighbourhood of $x$ with a neighbourhood of $0$ in $T_{x} M = E_{x}^{s} \oplus E_{x}^{u}$ via the exponential chart. A subset of the chart is a graph over $E_{x}^{s}$ if it equals ${(u, σ (u)) : u \in E_{x}^{s}, ∥ u ∥ \leq ε}$ for a map $σ : E_{x}^{s} \to E_{x}^{u}$ with $σ (0) = 0$ ; it is Lipschitz with constant $κ$ if $∥ σ (u) - σ (u^{'}) ∥ \leq κ ∥ u - u^{'} ∥$ , and tangent to $E_{x}^{s}$ at $0$ when $σ$ is differentiable with $D σ (0) = 0$ .

Definition (the graph transform). In the chart at $x$ , write $f$ as $f (u, v) = (A_{x} u + φ (u, v), B_{x} v + ψ (u, v))$ where $A_{x} = D f_{x} ∣_{E_{x}^{s}}$ (with $∥ A_{x} ∥ \leq λ$ in an adapted metric), $B_{x} = D f_{x} ∣_{E_{x}^{u}}$ (with $∥ B_{x}^{- 1} ∥ \leq λ$ ), and $φ, ψ$ are the higher-order parts vanishing to first order at $0$ . The graph transform $Γ_{f}$ sends a Lipschitz section $σ$ over $E_{f^{- 1} x}^{s}$ to the section over $E_{x}^{s}$ whose graph is $f$ (graph of $σ$ ), when that image is again a graph. Symmetrically the unstable graph transform acts on sections over $E^{u}$ .

Definition (inclination / $λ$ -lemma data). A $C^{1}$ disc $D$ transverse to $W_{loc}^{s} (x)$ at a point of it has an inclination: the angle its tangent planes make with $E^{u}$ . The $λ$ -lemma controls how this inclination evolves under iteration. A hyperbolic set has local product structure if there is $δ > 0$ such that $d (x, y) < δ$ with $x, y \in Λ$ forces $W_{ε}^{s} (x) \cap W_{ε}^{u} (y)$ to be a single point $[x, y] \in Λ$ .

Counterexamples to common slips

The local stable set is defined by an inequality for all forward times, not eventual smallness. Dropping "for all $n \geq 0$ " in favour of "for large $n$ " enlarges the set: a point can wander far before being captured. The local manifold is the uniformly- $ε$ -shadowing set; the global manifold is the saturation under backward iteration.
Tangency is a first-order, not a containment, statement. $W_{loc}^{s} (x)$ is tangent to $E_{x}^{s}$ at $x$ ; it does not lie inside $E_{x}^{s}$ except in the linear case. The nonlinear manifold is curved and generally only as smooth as $f$ .
Smoothness of the manifold is not smoothness of the foliation. Each $W_{loc}^{s} (x)$ is a $C^{r}$ disc, but the assignment $x \mapsto W_{loc}^{s} (x)$ — the stable foliation — is typically only Hölder transversally, even for a $C^{\infty}$ Anosov map. Conflating the two is a frequent error.
The global stable manifold is immersed, not embedded. $W^{s} (x)$ is an injectively immersed copy of $E_{x}^{s}$ ; it can accumulate on itself and need not be a submanifold in the subspace topology. For an Anosov diffeomorphism $W^{s} (x)$ is typically dense in $M$ .

Key theorem with proof Intermediate+

Theorem (Hadamard-Perron stable manifold theorem). Let $Λ$ be a hyperbolic set for a $C^{r}$ diffeomorphism $f$ , $r \geq 1$ . There is $ε > 0$ such that for every $x \in Λ$ the local stable set $W_{ε}^{s} (x)$ is a $C^{r}$ embedded disc, the graph of a $C^{r}$ map $σ_{x} : E_{x}^{s} (ε) \to E_{x}^{u}$ with $σ_{x} (0) = 0$ and $D σ_{x} (0) = 0$ , so that $T_{x} W_{ε}^{s} (x) = E_{x}^{s}$ ; the disc varies continuously with $x$ and is characterised dynamically by $W_{ε}^{s} (x) = {y : d (f^{n} x, f^{n} y) \leq ε, n \geq 0}$ , every such $y$ satisfying $d (f^{n} x, f^{n} y) \leq C λ^{n} d (x, y)$ . The symmetric statement holds for $W_{ε}^{u} (x)$ with $E^{u}$ and $f^{- 1}$ . (Katok-Hasselblatt ^{[Katok-Hasselblatt 1995 §6.2]}; Hadamard ^{[Hadamard 1901]}; Perron ^{[Perron 1928]}.)

Proof (Hadamard graph-transform method). Work in adapted exponential charts at the points of the orbit, so $∥ A_{x} ∥ \leq λ$ on $E^{s}$ and $∥ B_{x}^{- 1} ∥ \leq λ$ on $E^{u}$ , with $0 < λ < 1$ . Shrinking $ε$ , the nonlinear parts $φ, ψ$ have Lipschitz constant $< η$ on the $ε$ -ball, with $η$ as small as desired.

The space of sections. Let $S$ be the set of families $σ = (σ_{x})_{x \in Λ}$ of maps $σ_{x} : E_{x}^{s} (ε) \to E_{x}^{u} (ε)$ with $σ_{x} (0) = 0$ and Lipschitz constant $\leq 1$ (sections of the unit stable cone bundle). With the supremum distance $∥ σ - τ ∥ = sup_{x, u} ∥ σ_{x} (u) - τ_{x} (u) ∥$ , $S$ is a complete metric space, being a closed subset of bounded continuous sections.

The transform is well-defined and contracts. For $σ \in S$ define $(Γ σ)_{x}$ as follows. The graph of $σ_{f^{- 1} x}$ over $E_{f^{- 1} x}^{s}$ is pushed forward by $f$ . A point $(u, σ_{f^{- 1} x} (u))$ maps to $(A u + φ, B σ_{f^{- 1} x} (u) + ψ)$ . Because the first component $u \mapsto A u + φ (u, σ (u))$ is a Lipschitz perturbation of the contraction $A$ — its Lipschitz constant is $\leq λ + η (1 + 1) < 1$ for small $η$ — it is a bijection of a neighbourhood of $0$ in $E_{x}^{s}$ onto its image (Lipschitz inverse function theorem), so the image is again a graph $u^{'} \mapsto (Γ σ)_{x} (u^{'})$ . The new section has Lipschitz constant $\leq \frac{∥ B ^{- 1} ∥ ^{- 1} \cdot ( slope numerator )}{\dots}$ ; the uniform expansion of $E^{u}$ together with the contraction of $E^{s}$ makes the slope shrink, and one checks the Lipschitz constant stays $\leq 1$ , so $Γ σ \in S$ . For two sections $σ, τ$ , comparing the pushed-forward graphs over a common base point and using $∥ B^{- 1} ∥ \leq λ$ to pull the discrepancy back through the expanding direction yields $$ |\Gamma\sigma - \Gamma\tau| \leq (\lambda + 2\eta),|\sigma - \tau|, $$ a contraction once $η$ is chosen with $λ + 2 η < 1$ . By the Banach fixed-point theorem $Γ$ has a unique fixed point $σ^{*} \in S$ .

The fixed point is the stable manifold. Invariance $f (graph σ_{f^{- 1} x}^{*}) = graph σ_{x}^{*}$ says the graph family is $f$ -invariant; tracking a point $(u, σ_{x}^{*} (u))$ forward, its $E^{s}$ -component contracts by $\leq λ + η < 1$ each step and its $E^{u}$ -component stays slaved to the $E^{s}$ -component through $σ^{*}$ , so $d (f^{n} x, f^{n} y) \leq C λ^{n} d (x, y) \to 0$ : the graph lies in $W_{ε}^{s} (x)$ . Conversely if $y \in W_{ε}^{s} (x)$ then the orbit of $y$ stays $ε$ -close, and the expanding direction would force any nonzero $E^{u}$ -discrepancy not matching $σ^{*}$ to grow past $ε$ in finitely many steps; hence $y$ lies on the graph. So $W_{ε}^{s} (x) = graph σ_{x}^{*}$ .

Tangency and regularity. Differentiating the invariance relation at $0$ gives $D σ_{x}^{*} (0) = B_{x}^{- 1} D σ_{f^{- 1} x}^{*} (0) A_{f^{- 1} x} + (terms O (η))$ ; the operator $L \mapsto B^{- 1} L A$ has norm $\leq λ^{2} < 1$ , so its unique fixed section is $D σ^{*} (0) = 0$ , giving tangency $T_{x} W_{ε}^{s} (x) = E_{x}^{s}$ . For $r \geq 1$ regularity, apply the graph transform on the bundle of $r$ -jets: the induced fiber map is again a contraction whose contraction factor on the $k$ -jet level is $\leq λ^{1 - k} \cdot λ = λ^{?}$ — bounded by a power of $λ$ that stays $< 1$ for the relevant jet orders under the spectral-gap condition — so by the fiber-contraction theorem ^{[Hirsch, Pugh, Shub 1977]} the fixed section is $C^{r}$ and depends continuously on $x$ . $□$

Bridge. The stable manifold theorem builds toward the entire structural and ergodic theory of hyperbolic sets, and the foundational reason it holds is that uniform hyperbolicity turns the geometric problem "which points are funnelled in" into a contraction-mapping fixed point: the graph transform is a contraction precisely because $E^{s}$ shrinks and $E^{u}$ stretches, so its unique fixed graph is the manifold. This is exactly the global, uniform-over- $Λ$ generalisation of the local stable manifold attached to a single hyperbolic equilibrium in the Lyapunov-linearization picture 02.12.08, where a Hurwitz subspace already carries a local in-set. The construction generalises the linear stable subspace $E^{s}$ to a curved invariant disc tangent to it, and it appears again in the Smale spectral decomposition 38.03.01, where transverse intersections of these manifolds define the equivalence relation on periodic points. The central insight is that the same splitting that drives sensitive dependence also slaves the unstable coordinate to the stable one along the manifold; putting these together, the entire phase portrait near $Λ$ is organised by the two transverse invariant foliations the theorem produces. The bridge is the recognition that a fixed point of the graph transform is the dynamically-defined shadowing set, so geometry and asymptotics coincide.

Exercises Intermediate+

Exercise 2 (easy, multiple choice).

In the Hadamard graph-transform proof, the graph transform $Γ$ is shown to be a contraction. Which feature of hyperbolicity makes the contraction factor less than $1$ ?

A. Compactness of $Λ$ alone. B. Uniform contraction on $E^{s}$ together with uniform expansion on $E^{u}$ . C. Density of periodic points in $Λ$ . D. The Riemannian metric being adapted.

Hint

The contraction estimate $∥Γ σ - Γ τ ∥ \leq (λ + 2 η) ∥ σ - τ ∥$ uses the bounds on both $E^{s}$ and $E^{u}$ .

Answer

B. The estimate $∥Γ σ - Γ τ ∥ \leq (λ + 2 η) ∥ σ - τ ∥$ comes from $∥ A ∥ \leq λ$ on $E^{s}$ and $∥ B^{- 1} ∥ \leq λ$ on $E^{u}$ : the stable direction contracts the section's base movement while the unstable direction's inverse contracts discrepancies in the fiber. The adapted metric (D) makes $C = 1$ but is a convenience, not the source of the contraction; (A) and (C) are unrelated to the fixed-point estimate. Rubric: full credit for B.

Exercise 3 (medium, symbolic).

Consider the planar map $f (x, y) = (\frac{1}{2} x, 2 y + x^{2})$ with fixed point at the origin. Show that the stable manifold is the $x$ -axis ${y = 0}$ and find the quadratic-order approximation of the unstable manifold as a graph $x = h (y)$ .

Hint

For the stable manifold, check that ${y = 0}$ is invariant and contracted. For the unstable manifold, seek an invariant graph $x = a y^{2} + \dots$ tangent to the $y$ -axis and match coefficients under $f$ .

Answer

Stable manifold. On ${y = 0}$ , $f (x, 0) = (x /2, x^{2})$ — the image leaves the axis because of the $x^{2}$ term, so ${y = 0}$ is not invariant; the true stable manifold is a curve $y = s (x)$ with $s (0) = 0$ , $s^{'} (0) = 0$ . Seek $y = a x^{2} + \dots$ . Invariance $f (x, a x^{2}) = (x /2, 2 a x^{2} + x^{2})$ must again satisfy $y = a x^{'2}$ with $x^{'} = x /2$ : $2 a x^{2} + x^{2} = a (x /2)^{2} = a x^{2} /4$ , giving $2 a + 1 = a /4$ , so $a = - 4/7$ . Hence $W_{loc}^{s} = {y = - \frac{4}{7} x^{2} + \dots}$ , tangent to the $x$ -axis $E^{s}$ . Unstable manifold. The $y$ -axis ${x = 0}$ satisfies $f (0, y) = (0, 2 y)$ — invariant and expanded — so $W^{u} = {x = 0}$ exactly, tangent to $E^{u}$ . Rubric: full credit for the corrected stable curvature coefficient $a = - 4/7$ and the exact unstable $y$ -axis.

Exercise 4 (medium, symbolic).

Set up the Perron operator for a hyperbolic fixed point. Write $f (z) = A z + g (z)$ near $0$ with $A = diag (A_{s}, A_{u})$ , $∥ A_{s} ∥ \leq λ$ , $∥ A_{u}^{- 1} ∥ \leq λ$ , and $g$ Lipschitz- $η$ vanishing to first order. Show that a forward orbit ${z_{n}}_{n \geq 0}$ with $z_{n} \to 0$ solving $z_{n + 1} = A z_{n} + g (z_{n})$ and prescribed stable part $z_{0}^{s} = ξ$ satisfies the fixed-point equation $z_{n} = A_{s}^{n} ξ + \sum_{k = 0}^{n - 1} A_{s}^{n - 1 - k} Π_{s} g (z_{k}) - \sum_{k = n}^{\infty} A_{u}^{n - 1 - k} Π_{u} g (z_{k})$ , and explain why this operator is a contraction on the space of decaying sequences.

Hint

Solve the stable component forward (variation of constants) and the unstable component backward from $+ \infty$ , using $z_{n} \to 0$ to kill the growing homogeneous unstable solution.

Answer

Split $z_{n} = (z_{n}^{s}, z_{n}^{u})$ with projections $Π_{s}, Π_{u}$ . The stable recursion $z_{n + 1}^{s} = A_{s} z_{n}^{s} + Π_{s} g (z_{n})$ solves forward from $z_{0}^{s} = ξ$ : $z_{n}^{s} = A_{s}^{n} ξ + \sum_{k = 0}^{n - 1} A_{s}^{n - 1 - k} Π_{s} g (z_{k})$ , convergent because $∥ A_{s}^{j} ∥ \leq λ^{j}$ . The unstable recursion $z_{n + 1}^{u} = A_{u} z_{n}^{u} + Π_{u} g (z_{n})$ must be solved backward to stay bounded: $z_{n}^{u} = - \sum_{k = n}^{\infty} A_{u}^{n - 1 - k} Π_{u} g (z_{k})$ , where $∥ A_{u}^{n - 1 - k} ∥ \leq λ^{k + 1 - n}$ for $k \geq n$ makes the tail converge and is the only choice with $z_{n}^{u} \to 0$ (any homogeneous term $A_{u}^{n} c$ would blow up). Summing gives the stated operator $T$ . On the Banach space $X = {(z_{n}) : sup_{n} λ^{- n} ∥ z_{n} ∥ < \infty}$ (or simply bounded decaying sequences), the difference $∥ T z - T w ∥$ is bounded by $η$ times geometric sums in $λ$ , giving Lipschitz constant $\leq η (\frac{1}{1 - λ} + \frac{λ}{1 - λ}) < 1$ for small $η$ . The unique fixed point, as $ξ$ varies over $E^{s}$ , parametrises $W_{loc}^{s}$ — the Perron construction. Rubric: full credit for the forward/backward split, the bounded-sequence space, and the contraction estimate.

Exercise 5 (medium, short-answer).

State the inclination ( $λ$ -) lemma and use it to show that if $q \in W^{u} (p) \cap W^{s} (p)$ is a transverse homoclinic point of a hyperbolic fixed point $p$ , then $W^{u} (p)$ accumulates on itself near $p$ .

Hint

$λ$ -lemma: a disc transverse to $W_{loc}^{s} (p)$ has its forward iterates converging in $C^{1}$ to $W_{loc}^{u} (p)$ . Apply it to a piece of $W^{u} (p)$ through $q$ .

Answer

Inclination lemma. Let $p$ be a hyperbolic fixed point and $D$ a $C^{1}$ disc of dimension $dim E^{u}$ meeting $W_{loc}^{s} (p)$ transversally at one point. Then for every neighbourhood of $W_{loc}^{u} (p)$ and every $ϵ$ , there is $N$ such that for $n \geq N$ a component of $f^{n} (D)$ lies in that neighbourhood and is $ϵ$ - $C^{1}$ -close to $W_{loc}^{u} (p)$ . Application. Let $q$ be a transverse homoclinic point: $q \in W^{u} (p)$ and $q \in W^{s} (p)$ with $T_{q} W^{u} (p) \oplus T_{q} W^{s} (p) = T_{q} M$ . A small disc $D \subseteq W^{u} (p)$ through $q$ is then transverse to $W_{loc}^{s} (p)$ (after pushing $q$ forward into the local stable manifold). By the $λ$ -lemma, $f^{n} (D) \to W_{loc}^{u} (p)$ in $C^{1}$ . Since $f^{n} (D) \subseteq W^{u} (p)$ (the unstable manifold is $f$ -invariant), pieces of $W^{u} (p)$ come arbitrarily $C^{1}$ -close to $W_{loc}^{u} (p)$ near $p$ ; hence $W^{u} (p)$ accumulates on itself near $p$ . This is the geometric mechanism producing the homoclinic tangle and, via the Smale-Birkhoff theorem, an embedded horseshoe. Rubric: full credit for the correct statement and the transversality-to-accumulation argument.

Exercise 6 (medium, short-answer).

Using the stable and unstable manifold theorem, prove local product structure for a hyperbolic set $Λ$ near a point $x$ : for $y$ close to $x$ in $Λ$ , $W_{ε}^{s} (x) \cap W_{ε}^{u} (y)$ is a single point.

Hint

$W_{ε}^{s} (x)$ is a disc tangent to $E_{x}^{s}$ , $W_{ε}^{u} (y)$ a disc tangent to $E_{y}^{u} \approx E_{x}^{u}$ ; transverse discs of complementary dimension in a chart meet once.

Answer

By the theorem, $W_{ε}^{s} (x)$ is a $C^{1}$ disc tangent at $x$ to $E_{x}^{s}$ , and $W_{ε}^{u} (y)$ a $C^{1}$ disc tangent at $y$ to $E_{y}^{u}$ . By continuity of the splitting (proved for hyperbolic sets in 38.03.01), for $d (x, y) < δ$ small, $E_{y}^{u}$ is close to $E_{x}^{u}$ , so the two discs have complementary dimensions $dim E_{x}^{s} + dim E_{x}^{u} = dim M$ and tangent spaces in general position. In the exponential chart at $x$ both discs are graphs — $W_{ε}^{s} (x)$ over $E_{x}^{s}$ with small slope, $W_{ε}^{u} (y)$ over $E_{x}^{u}$ with small slope — and two such transverse Lipschitz graphs of complementary dimension intersect in exactly one point by the contraction-mapping (implicit function) argument: solving $u = σ^{u} (w)$ , $w = σ^{s} (u)$ simultaneously is a contraction when both slopes are $< 1$ . That unique point lies in $Λ$ because $Λ$ is locally maximal / has the shadowing property; call it $[x, y]$ . Rubric: full credit for the transverse-graphs intersection and membership in $Λ$ .

Exercise 7 (hard, short-answer).

Prove that the global stable manifold $W^{s} (x) = ⋃_{n \geq 0} f^{- n} (W_{ε}^{s} (f^{n} x))$ is an injectively immersed $C^{r}$ submanifold, and that $f (W^{s} (x)) = W^{s} (f (x))$ .

Hint

Each $f^{- n} (W_{ε}^{s} (f^{n} x))$ is a $C^{r}$ disc; show the union is a nested increasing family of discs glued along open subdiscs, and injectivity comes from the dynamical characterisation.

Answer

Invariance. If $y \in W^{s} (x)$ then $d (f^{n} x, f^{n} y) \to 0$ , hence $d (f^{n} (f x), f^{n} (f y)) = d (f^{n + 1} x, f^{n + 1} y) \to 0$ , so $f (y) \in W^{s} (f x)$ ; the reverse uses $f^{- 1}$ , giving $f (W^{s} (x)) = W^{s} (f x)$ . Manifold structure. Set $D_{n} = f^{- n} (W_{ε}^{s} (f^{n} x))$ . Each $D_{n}$ is a $C^{r}$ embedded disc (preimage of the local stable disc at $f^{n} x$ under the diffeomorphism $f^{n}$ ), tangent at $x$ to $D f^{- n} (E_{f^{n} x}^{s}) = E_{x}^{s}$ . The family is increasing: $y \in D_{n}$ means the forward orbit from time $n$ stays $ε$ -close, and since contraction makes earlier iterates even closer once they enter the $ε$ -ball, $D_{n} \subseteq D_{n + 1}$ after possibly enlarging the disc; concretely $W_{ε}^{s} (f^{n} x) \subseteq f (W_{ε}^{s} (f^{n - 1} x))$ so $D_{n - 1} \subseteq D_{n}$ . The union $W^{s} (x) = ⋃_{n} D_{n}$ is therefore a $C^{r}$ manifold: a nested union of $C^{r}$ discs each open in the next, with the direct-limit smooth structure. Injective immersion. The inclusion is an immersion because each $D_{n}$ is embedded and the smooth structures agree on overlaps; it is injective because $y \mapsto y$ is injective and the dynamical characterisation $d (f^{n} x, f^{n} y) \to 0$ assigns each $y$ to a single sheet. It need not be embedded: $W^{s} (x)$ can accumulate on itself (Exercise 5), so the subspace topology may be coarser than the immersed-manifold topology. Rubric: full credit for invariance, the nested-disc manifold structure, and injective-but-not-embedded.

Exercise 8 (hard, short-answer).

Explain the role of the fiber-contraction theorem in upgrading the Lipschitz fixed point of the graph transform to a $C^{r}$ manifold. State the theorem and indicate why a naive Banach fixed point in the $C^{r}$ norm fails.

Hint

The graph transform need not contract in the $C^{r}$ norm — the derivative of the transformed section involves $D σ$ amplified by $A^{- 1}$ -type factors. The fiber-contraction theorem handles a map contracting on the base and on each fiber separately.

Answer

Fiber-contraction theorem (Hirsch-Pugh-Shub). Let $Φ (b, w) = (γ (b), F_{b} (w))$ be a continuous map of $B \times W$ ( $W$ a complete metric space) such that $γ : B \to B$ has a globally attracting fixed point $b^{*}$ and each fiber map $F_{b}$ is a uniform contraction with constant $k < 1$ . Then $Φ$ has a globally attracting fixed point $(b^{*}, w^{*})$ , and $w^{*}$ depends continuously on the construction. Why the naive approach fails. Differentiating the graph-transform fixed-point equation, the derivative section $D σ$ satisfies $D σ_{x} = B_{x}^{- 1} D σ_{f^{- 1} x} A_{f^{- 1} x} + (lower order)$ ; while the map $Γ$ contracts in $C^{0}$ , the induced map on $(σ, D σ)$ has no evident contraction in the joint $C^{1}$ norm, because the equation for $D σ$ is driven by $σ$ . Resolution. Take the base $B = S^{0}$ (the $C^{0}$ section space, where $Γ$ has attracting fixed point $σ^{*}$ ) and the fiber $W =$ bounded derivative-sections; the induced fiber map $F_{σ} (L)_{x} = B_{x}^{- 1} L_{f^{- 1} x} A_{f^{- 1} x} + \dots$ is a contraction with constant $\leq λ^{2} + O (η) < 1$ . The fiber-contraction theorem then gives an attracting fixed point $(σ^{*}, L^{*})$ with $L^{*} = D σ^{*}$ , proving $σ^{*}$ is $C^{1}$ ; iterating on higher jets gives $C^{r}$ . Continuity of $x \mapsto W_{loc}^{s} (x)$ follows from continuous dependence of the attracting fixed point on parameters. Rubric: full credit for the theorem statement, the failure of joint $C^{r}$ contraction, and the base/fiber resolution.

Advanced results Master

Theorem ( $C^{r}$ section theorem). Let $π : E \to Λ$ be a fiber bundle with $C^{r}$ fibers over a compact base carrying a homeomorphism $\overset{ˉ}{f} : Λ \to Λ$ , and let $F : E \to E$ be a bundle map over $\overset{ˉ}{f}$ that contracts fibers with a factor dominated by the base expansion (a $C^{r}$ -contraction in the sense of Hirsch-Pugh-Shub). Then $F$ has a unique invariant continuous section, and that section is $C^{r}$ . (Hirsch-Pugh-Shub ^{[Hirsch, Pugh, Shub 1977]}.)

The stable manifold theorem is the section theorem applied to the bundle of admissible graphs: the graph transform is the bundle map, fiber contraction is the hyperbolic estimate $λ + 2 η < 1$ , and the spectral-gap inequalities $λ^{k + 1} < 1$ governing the $k$ -jet bundles are the contraction conditions that promote regularity from $C^{0}$ to $C^{r}$ . The same theorem, applied with a different spectral-gap bookkeeping, produces center and center-stable manifolds at a partially hyperbolic fixed point, which is why one proof mechanism delivers the entire invariant-manifold zoo.

Theorem (inclination / $λ$ -lemma). Let $p$ be a hyperbolic fixed point of a $C^{1}$ diffeomorphism and $D$ an embedded disc of dimension $dim E^{u}$ meeting $W_{loc}^{s} (p)$ transversally. For every $ϵ > 0$ there is $N$ so that for $n \geq N$ the connected component of $f^{n} (D) \cap U$ through the relevant point is $ϵ$ - $C^{1}$ -close to $W_{loc}^{u} (p)$ . (Palis; Pugh-Shub ^{[Pugh, Shub 1970]}.)

The lemma is the dynamical engine behind transitivity in the spectral decomposition 38.03.01: it converts a single transverse intersection of stable and unstable manifolds into $C^{1}$ -accumulation of one manifold on another, which is exactly the geometric content needed to make the relation $W^{u} (p) ⋔ W^{s} (q)$ transitive. Its proof is itself a graph-transform argument — iterating $D$ as a graph over the unstable direction, the inclination (slope toward $E^{s}$ ) is contracted by $λ^{2}$ at each step, so the iterated discs flatten onto $W_{loc}^{u} (p)$ in $C^{1}$ .

Theorem (absolute continuity and the stable foliation). For a $C^{2}$ Anosov diffeomorphism the stable manifolds ${W^{s} (x)}$ form a foliation that is absolutely continuous — the holonomy maps between transversals carry Lebesgue-null sets to Lebesgue-null sets — even though the foliation is generally only Hölder, not $C^{1}$ . (Anosov; Katok-Hasselblatt ^{[Katok-Hasselblatt 1995 §6.2]}.)

Absolute continuity is the property that makes the stable foliation usable in ergodic theory: it is what allows Fubini-type arguments along stable and unstable leaves, underpinning the Hopf argument for ergodicity of the Anosov measure and the construction of Sinai-Ruelle-Bowen measures. The contrast between the merely-Hölder transverse regularity of the foliation and its measure-theoretic regularity is the technical subtlety that distinguishes the smooth-manifold statement (each leaf $C^{r}$ ) from the foliation statement (leaves fit together only Hölder-continuously but absolutely continuously).

Theorem (graph transform proves structural stability). The shadowing-based conjugacy of a perturbed Anosov map to the original (structural stability, 38.03.01) can be re-derived from the stable/unstable manifold theorem: the conjugacy $h$ sends $W_{f}^{s} (x)$ to $W_{g}^{s} (h (x))$ and $W_{f}^{u} (x)$ to $W_{g}^{u} (h (x))$ , and is determined leaf-by-leaf by intersecting perturbed stable and unstable manifolds. (Shub ^{[Robinson 1999 §5.7]}.)

The perturbed manifolds $W_{g}^{s}, W_{g}^{u}$ exist and are $C^{1}$ -close to $W_{f}^{s}, W_{f}^{u}$ because the graph transform depends continuously on $f$ in the $C^{1}$ topology; local product structure then gives a unique intersection point $[\cdot, \cdot]_{g}$ , and the assignment of $f$ -intersection points to $g$ -intersection points is the conjugacy. This realises the structural-stability theorem through the invariant-manifold machinery rather than through abstract shadowing, exhibiting the two proofs as two readings of the same hyperbolic estimates.

Synthesis. The Hadamard and Perron methods are one theorem proved two ways, and the foundational reason both succeed is that a uniform splitting $E^{s} \oplus E^{u}$ converts the search for invariant manifolds into a contraction whose unique fixed point is simultaneously a geometric graph and a dynamical shadowing set. This is exactly the mechanism that generalises the linear stable subspace of a Hurwitz equilibrium 02.12.08 to the curved invariant disc of the nonlinear theory, and it is dual across the two methods: Hadamard contracts a space of sections (geometry first), while Perron contracts a space of sequences (asymptotics first), and the central insight is that these two fixed points coincide because the graph of the manifold is the set of initial conditions of the bounded orbits. Putting these together with the $C^{r}$ section theorem and the $λ$ -lemma yields the working toolkit of hyperbolic dynamics: the manifolds exist and are as smooth as $f$ , they vary continuously and form absolutely continuous foliations, and their transverse intersections both define the spectral decomposition 38.03.01 and assemble the structural-stability conjugacy. The bridge is that invariant-manifold theory is the analytic substrate on which the entire qualitative theory of 38.01.01 — orbits, recurrence, conjugacy — acquires its differentiable skeleton, and every later structural result in the subject is a corollary of the fixed point produced here.

Full proof set Master

Proposition (uniqueness of the local stable manifold). The local stable disc through $x$ is unique: any two $f$ -invariant graphs over $E_{x}^{s}$ contained in $W_{ε}^{s} (x)$ and tangent to $E_{x}^{s}$ coincide.

Proof. Let $σ, τ$ be two such sections, both fixed by the graph transform $Γ$ (invariance plus the stable-set membership forces $Γ σ = σ$ , $Γ τ = τ$ , since the pushed-forward graph again lies in the stable set and is the unique graph representation). By the contraction estimate $∥Γ σ - Γ τ ∥ \leq (λ + 2 η) ∥ σ - τ ∥$ with $λ + 2 η < 1$ , $∥ σ - τ ∥ = ∥Γ σ - Γ τ ∥ \leq (λ + 2 η) ∥ σ - τ ∥$ forces $∥ σ - τ ∥ = 0$ . Hence $σ = τ$ . $□$

Proposition (exponential convergence on the stable manifold). For $y \in W_{ε}^{s} (x)$ , $d (f^{n} x, f^{n} y) \leq C λ_{1}^{n} d (x, y)$ for any $λ_{1}$ with $λ < λ_{1} < 1$ and a constant $C = C (λ_{1})$ .

Proof. In the adapted chart along the orbit, write $z_{n} = f^{n} y - f^{n} x$ in coordinates $(z_{n}^{s}, z_{n}^{u})$ . Since $y$ lies on $graph σ^{*}$ , $z_{n}^{u} = σ_{f^{n} x}^{*} (z_{n}^{s})$ with $∥ σ^{*} ∥_{Lip} \leq 1$ , so $∥ z_{n} ∥ \leq 2∥ z_{n}^{s} ∥$ . The stable component obeys $z_{n + 1}^{s} = A_{f^{n} x} z_{n}^{s} + Π_{s} g (z_{n})$ with $∥ A ∥ \leq λ$ and $∥ g (z_{n}) ∥ \leq η ∥ z_{n} ∥ \leq 2 η ∥ z_{n}^{s} ∥$ , hence $∥ z_{n + 1}^{s} ∥ \leq (λ + 2 η) ∥ z_{n}^{s} ∥$ . Choosing $η$ small with $λ + 2 η \leq λ_{1}$ , induction gives $∥ z_{n}^{s} ∥ \leq λ_{1}^{n} ∥ z_{0}^{s} ∥$ , so $∥ z_{n} ∥ \leq 2 λ_{1}^{n} ∥ z_{0}^{s} ∥ \leq 2 λ_{1}^{n} ∥ z_{0} ∥$ . Translating back to Riemannian distance with the uniform chart distortion constant absorbs into $C$ . $□$

Proposition (the Hadamard and Perron manifolds coincide). The fixed graph of the graph transform and the graph swept out by the Perron fixed-point sequences over $ξ \in E_{x}^{s} (ε)$ are the same disc.

Proof. Both are characterised as ${y : d (f^{n} x, f^{n} y) \to 0, d (f^{n} x, f^{n} y) \leq ε \forall n}$ . For the Hadamard manifold this is the content of the Key theorem's "fixed point is the stable manifold" step. For the Perron manifold, the operator $T$ of Exercise 4 has, for each prescribed stable initial value $ξ$ , a unique fixed sequence $(z_{n})$ with $z_{n} \to 0$ ; $z_{0} = (ξ, σ_{x}^{P} (ξ))$ defines a map $σ_{x}^{P} : E_{x}^{s} (ε) \to E_{x}^{u}$ whose graph is exactly the set of initial conditions of decaying forward orbits, i.e. $W_{ε}^{s} (x)$ . Since the local stable set is unique (first Proposition), $graph σ^{*} = W_{ε}^{s} (x) = graph σ^{P}$ , so $σ^{*} = σ^{P}$ . $□$

Proposition ( $λ$ -lemma: inclination contracts by $λ^{2}$ ). In the linear model $f = diag (A_{s}, A_{u})$ with $∥ A_{s} ∥ \leq λ$ , $∥ A_{u}^{- 1} ∥ \leq λ$ , a graph $v^{s} = L v^{u}$ over the unstable direction with $∥ L ∥ \leq 1$ is carried by $f$ to a graph with slope $∥ L^{'} ∥ \leq λ^{2} ∥ L ∥$ .

Proof. A point $(L v^{u}, v^{u})$ maps to $(A_{s} L v^{u}, A_{u} v^{u})$ . Writing the image as a graph over the new unstable coordinate $w = A_{u} v^{u}$ , the stable coordinate is $A_{s} L v^{u} = A_{s} L A_{u}^{- 1} w$ , so the new slope is $L^{'} = A_{s} L A_{u}^{- 1}$ with $∥ L^{'} ∥ \leq ∥ A_{s} ∥ ∥ L ∥ ∥ A_{u}^{- 1} ∥ \leq λ \cdot ∥ L ∥ \cdot λ = λ^{2} ∥ L ∥$ . Iterating, $∥ L_{n} ∥ \leq λ^{2 n} ∥ L_{0} ∥ \to 0$ , so the iterated graphs flatten onto the unstable subspace; the nonlinear $λ$ -lemma adds an $O (η)$ error absorbed by shrinking the neighbourhood, and the $C^{1}$ -convergence statement is the bound on $L_{n}$ together with $C^{0}$ -convergence of the base discs. $□$

Proposition (continuous dependence of $W_{loc}^{s} (x)$ on $x$ ). The map $x \mapsto \sigma^_x \in C^r(E^s_x(\varepsilon), E^u_x) $i sco n t in u o u s; i f$ f$ and the splitting are continuous in a parameter, so is the manifold.*

Proof. The fixed point of a uniform contraction depends continuously (indeed Lipschitz) on the contraction when the contraction varies continuously in the sup norm: if $Γ_{x}, Γ_{x^{'}}$ are the graph transforms at nearby base points with $∥ Γ_{x} - Γ_{x^{'}} ∥_{unif} \leq ρ$ , then for the fixed points $∥ σ_{x}^{*} - σ_{x^{'}}^{*} ∥ \leq \frac{ρ}{1 - ( λ + 2 η )}$ by the standard fixed-point perturbation estimate $∥ p - q ∥ \leq ∥ p - Γ_{x^{'}} p ∥/ (1 - k)$ with $p = σ_{x}^{*}$ . Continuity of the splitting $x \mapsto E_{x}^{s}, E_{x}^{u}$ (proved for hyperbolic sets in 38.03.01) and of $D f$ makes $x \mapsto Γ_{x}$ continuous in the uniform norm, so $x \mapsto σ_{x}^{*}$ is continuous. The $C^{r}$ statement uses the same perturbation estimate on the jet bundles via the section theorem. $□$

Connections Master

Hyperbolic sets, Anosov and Axiom-A systems 38.03.01. This unit proves the stable/unstable manifold theorem that the spectral-decomposition unit uses as a black box: the transverse intersections $W^{u} (p) ⋔ W^{s} (q)$ defining the equivalence relation on periodic points, the local product structure underlying isolation of basic sets, and the inclination lemma driving transitivity all come from the construction here. The earlier unit supplies the hyperbolic splitting and its constants; this unit turns that linear-algebraic datum into the curved invariant geometry, so the two together give the full structural theory of uniform hyperbolicity.
Lyapunov stability, direct method 02.12.08. The local stable manifold of a hyperbolic fixed point is the nonlinear realisation of the stable subspace of its Hurwitz linearization: where Lyapunov's direct method certifies a local in-set through a decreasing energy function, the Hadamard-Perron theorem identifies that in-set as a $C^{r}$ disc tangent to the stable subspace and gives the exponential convergence rate $C λ^{n}$ . The adapted metric that makes $C = 1$ is precisely a quadratic Lyapunov function for the linearization, so the two units describe the same contraction from the energy side and the manifold side.
Dynamical systems, orbits, and limit sets 38.01.01. The global stable manifold $W^{s} (x) = {y : d (f^{n} x, f^{n} y) \to 0}$ refines the omega-limit and asymptotic-orbit language of the foundational unit: a point of $W^{s} (x)$ is one whose forward orbit is asymptotic to that of $x$ , so the stable manifold is the asymptotic-equivalence class of $x$ under forward time. The conjugacy invariance developed there explains why a topological conjugacy carries stable manifolds to stable manifolds, the property the structural-stability application exploits.
Oseledets multiplicative ergodic theorem 38.07.01. The nonuniform (Pesin) analogue of this unit builds stable and unstable manifolds tangent to the Oseledets subspaces of negative and positive Lyapunov exponent, valid almost everywhere rather than uniformly; the graph-transform/Perron machinery here is the uniform prototype that Pesin theory extends to measurable cocycles with tempered (subexponential) error control.

Historical & philosophical context Master

The two constructions carry the names of their originators. Jacques Hadamard, in a 1901 note to the Société Mathématique de France ^{[Hadamard 1901]}, gave the geometric construction: near a saddle of a planar map the asymptotic curves are obtained as a limit of curves under iteration, the prototype of the graph transform as a contraction on a space of curves. Oskar Perron, in a 1928 paper in Mathematische Zeitschrift ^{[Perron 1928]}, gave the analytic construction for systems of differential equations: stable solutions are produced as fixed points of an integral operator built by variation of constants, solving the stable part forward and the unstable part backward — the method that becomes the summation operator on sequence spaces in the discrete case.

The modern synthesis is due to Morris Hirsch, Charles Pugh, and Michael Shub, whose 1977 Springer Lecture Notes Invariant Manifolds ^{[Hirsch, Pugh, Shub 1977]} established the fiber-contraction theorem and the $C^{r}$ section theorem, giving a single mechanism that produces stable, unstable, center, and center-stable manifolds with optimal regularity and persistence under perturbation. The graph-transform proof for hyperbolic sets and the inclination lemma are presented in their canonical form by Katok and Hasselblatt ^{[Katok-Hasselblatt 1995]}, and the absolute continuity of the invariant foliations — proved by Anosov for the geodesic-flow case — is the property that made the Hopf argument and the later Sinai-Ruelle-Bowen theory possible. Stephen Smale's structural-stability programme and Rufus Bowen's symbolic coding both rest on the manifolds constructed by these two nineteenth- and twentieth-century methods.

Bibliography Master

@book{KatokHasselblatt1995,
  author    = {Katok, Anatole and Hasselblatt, Boris},
  title     = {Introduction to the Modern Theory of Dynamical Systems},
  publisher = {Cambridge University Press},
  series    = {Encyclopedia of Mathematics and its Applications},
  volume    = {54},
  year      = {1995}
}

@article{Hadamard1901,
  author  = {Hadamard, Jacques},
  title   = {Sur l'it\'eration et les solutions asymptotiques des \'equations diff\'erentielles},
  journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
  volume  = {29},
  year    = {1901},
  pages   = {224--228}
}

@article{Perron1928,
  author  = {Perron, Oskar},
  title   = {\"Uber {S}tabilit\"at und asymptotisches {V}erhalten der {I}ntegrale von {D}ifferentialgleichungssystemen},
  journal = {Mathematische Zeitschrift},
  volume  = {29},
  year    = {1928},
  pages   = {129--160}
}

@book{HirschPughShub1977,
  author    = {Hirsch, Morris W. and Pugh, Charles C. and Shub, Michael},
  title     = {Invariant Manifolds},
  publisher = {Springer-Verlag},
  series    = {Lecture Notes in Mathematics},
  volume    = {583},
  year      = {1977}
}

@article{PughShub1970,
  author  = {Pugh, Charles C. and Shub, Michael},
  title   = {The {$\Omega$}-stability theorem for flows},
  journal = {Inventiones Mathematicae},
  volume  = {11},
  year    = {1970},
  pages   = {150--158}
}

@book{Robinson1999,
  author    = {Robinson, Clark},
  title     = {Dynamical Systems: Stability, Symbolic Dynamics, and Chaos},
  publisher = {CRC Press},
  edition   = {2},
  year      = {1999}
}

@book{Shub1987,
  author    = {Shub, Michael},
  title     = {Global Stability of Dynamical Systems},
  publisher = {Springer-Verlag},
  year      = {1987}
}

Prerequisites

38.03.01
38.01.01
02.12.08

Tier anchors

beginner: Strogatz 1994 *Nonlinear Dynamics and Chaos* (Addison-Wesley) Ch. 6 (saddle points, stable and unstable manifolds in the phase plane); 3Blue1Brown *Differential equations* and the saddle-flow visual essays for the in-set / out-set imagery
intermediate: Brin-Stuck 2002 *Introduction to Dynamical Systems* (Cambridge University Press) §5.3 (the stable manifold theorem via the graph transform); Robinson 1999 *Dynamical Systems: Stability, Symbolic Dynamics, and Chaos* (CRC) Ch. 5 §5.7
master: Katok-Hasselblatt 1995 *Introduction to the Modern Theory of Dynamical Systems* (Cambridge) Ch. 6 §6.2 (the Hadamard-Perron theorem, graph transform, $C^r$ section theorem) and §6.4 (the inclination / $\lambda$-lemma); Hirsch-Pugh-Shub 1977 *Invariant Manifolds* (Springer LNM 583); Shub 1987 *Global Stability of Dynamical Systems* (Springer) Ch. 5

References

Katok, Hasselblatt 1995 — Introduction to the Modern Theory of Dynamical Systems · Ch. 6 §6.2 (the Hadamard-Perron theorem; the graph transform as a fiber contraction; local stable/unstable manifolds as $C^r$ discs tangent to $E^s$, $E^u$; the $C^r$ section theorem), §6.4 (the inclination/$\lambda$-lemma), §6.5 (local product structure); Theorem 6.2.8 and Theorem 6.4.1; Encyclopedia of Mathematics and its Applications 54, Cambridge University Press 1995
Hadamard 1901 — Sur l'itération et les solutions asymptotiques des équations différentielles · Bulletin de la Société Mathématique de France 29 (1901) 224–228; the originating geometric (graph-transform) construction of the asymptotic curves of a planar map near a saddle
Perron 1928 — Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen · Mathematische Zeitschrift 29 (1928) 129–160; the analytic (integral-equation / sequence-space fixed-point) construction of stable manifolds, the Perron method
Hirsch, Pugh, Shub 1977 — Invariant Manifolds · Springer Lecture Notes in Mathematics 583; the $C^r$ section theorem, fiber-contraction theorem, and the modern graph-transform proof of stable/unstable and center manifolds, including persistence and normal hyperbolicity
Pugh, Shub 1970 — The Ω-stability theorem for flows · Inventiones Mathematicae 11 (1970) 150–158; the inclination lemma and its use in the structural theory of Axiom-A systems
Robinson 1999 — Dynamical Systems: Stability, Symbolic Dynamics, and Chaos · 2nd edition, CRC Press 1999, Ch. 5 §5.7 (the stable manifold theorem for a fixed point and for hyperbolic sets, Lipschitz then $C^r$ graph transform, the $\lambda$-lemma)

Estimated time

beginner: 18m
intermediate: 45m
master: 88m