38.03.01 · dynamics / hyperbolicity-structural-stability

Hyperbolic Sets, Anosov and Axiom-A Systems, and the Smale Spectral Decomposition

shipped3 tiersLean: none

Anchor (Master): Katok-Hasselblatt 1995 *Introduction to the Modern Theory of Dynamical Systems* (Cambridge) Ch. 6 (hyperbolic sets), Ch. 17–18 (Anosov systems, Axiom A, the spectral decomposition); Shub 1987 *Global Stability of Dynamical Systems* (Springer); Smale 1967 *Differentiable dynamical systems* (Bull. Amer. Math. Soc. 73) — the originating survey of Axiom A, basic sets, and the spectral decomposition; Bowen 1975 *Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms* (Springer LNM 470)

Intuition Beginner

Picture stretching a sheet of dough between your hands: at every point, the dough is being pulled apart in one direction and squeezed together in the other. A hyperbolic dynamical system does exactly this to the space it acts on — at every point of a special invariant set, there is one bundle of directions the rule stretches and another bundle it shrinks, and these two roles never swap. The stretching is what makes nearby trajectories fly apart, the signature of chaos. The shrinking is what pins long-term behaviour onto a thin skeleton.

Why insist that the stretching and shrinking be uniform — bounded away from the no-change rate everywhere, all at once? Because that uniformity is what makes the chaos sturdy. A system whose hyperbolicity is uniform keeps its qualitative behaviour even if you bump the rule a little: the periodic orbits, the tangles, the whole pattern survive the perturbation. This robustness is the prize. It turns "this particular map is chaotic" into "every nearby map is chaotic in the same way".

When the entire space is hyperbolic, the system is called Anosov, and the cat map — a linear shuffle of a doughnut that stretches along one slanted direction and compresses along another — is the model. Smale's great discovery was that even when only part of the space is hyperbolic, the recurrent core breaks cleanly into finitely many self-contained chaotic pieces. That clean break is the spectral decomposition, and it is the organising map of the whole subject.

Visual Beginner

Imagine a single point sitting in the space, with a small cross drawn through it: one arm of the cross points along the stretching direction, the other along the shrinking direction. Apply the rule once. The cross at the new point has a longer stretching arm and a shorter shrinking arm. Apply it again and again, and the stretching arm grows without bound while the shrinking arm collapses toward nothing. Every point of a hyperbolic set carries such a cross, and the rule slides each cross to the next point while lengthening one arm and shortening the other.

A small table fixes the vocabulary.

name	what the rule does to it	everyday image
stable direction	shrinks lengths, uniformly	dough squeezed together
unstable direction	stretches lengths, uniformly	dough pulled apart
Anosov system	the whole space is hyperbolic	the cat map on a doughnut
basic set	one self-contained chaotic piece	one tangle in the spectral decomposition

Worked example Beginner

Take the cat map: the rule on a doughnut surface (a square with opposite edges glued) given by the matrix that sends a position $(x, y)$ to $(2 x + y, x + y)$ , then wraps the result back into the unit square. We check that one direction stretches and another shrinks.

Step 1. Find the special directions. The matrix has two slopes that it leaves pointing the same way, only rescaled. Solving, the stretching slope is about $0.618$ and the shrinking slope is about $- 1.618$ . Call the stretching direction $u$ and the shrinking direction $s$ .

Step 2. Find the rescaling factors. Along $u$ the rule multiplies lengths by $λ = (3 + 5) /2 \approx 2.618$ . Along $s$ it multiplies lengths by $1/ λ \approx 0.382$ .

Step 3. Track a tiny arrow pointing along $u$ , of length $1$ . After one step it has length $2.618$ . After two steps, $2.61 8^{2} \approx 6.85$ . After five steps, about $123$ . It grows without bound.

Step 4. Track a tiny arrow pointing along $s$ , of length $1$ . After one step it has length $0.382$ . After two steps $0.146$ , after five steps about $0.008$ . It collapses toward zero.

What this tells us: the same rule, at the very same point, pulls one direction apart by a fixed factor bigger than one and squeezes the perpendicular-in-spirit direction by a fixed factor below one, forever. Because these factors never depend on where you are on the doughnut, the whole surface is hyperbolic — this is what makes the cat map the model Anosov system, and the uniform factors $λ$ and $1/ λ$ are the reason its chaos survives small changes to the rule.

Check your understanding Beginner

Exercise (easy, multiple choice).

In a hyperbolic set, the rule splits the directions at each point into two kinds. Which pair correctly names them?

A. Fast directions and slow directions. B. Stable (shrinking) directions and unstable (stretching) directions. C. Forward directions and backward directions. D. Periodic directions and wandering directions.

Hint

One bundle of directions has its lengths squeezed toward zero by repeated application; the other has its lengths blown up.

Answer

B. At each point the tangent directions split into a stable bundle, whose vectors are uniformly contracted by forward iteration, and an unstable bundle, whose vectors are uniformly expanded. The two bundles together fill out all directions, and the rule never lets a vector switch which bundle it belongs to. The other options name distinctions that do not capture the stretch-versus-shrink dichotomy at the heart of hyperbolicity.

Formal definition Intermediate+

Let $M$ be a smooth Riemannian manifold and $f : M \to M$ a $C^{1}$ diffeomorphism. Write $D f_{x} : T_{x} M \to T_{f (x)} M$ for the derivative and $∥ \cdot ∥$ for the norm induced by the Riemannian metric.

Definition (hyperbolic set). A compact $f$ -invariant set $Λ \subseteq M$ (so $f (Λ) = Λ$ ) is hyperbolic if the restricted tangent bundle admits a continuous $D f$ -invariant splitting $$ T_\Lambda M = E^s \oplus E^u, \qquad Df_x(E^s_x) = E^s_{f(x)}, \quad Df_x(E^u_x) = E^u_{f(x)}, $$ together with constants $C \geq 1$ and $0 < λ < 1$ , independent of the point, such that for every $x \in Λ$ and every $n \geq 0$ , $$ |Df^n_x v| \leq C \lambda^n |v| \ \ (v \in E^s_x), \qquad |Df^{-n}_x v| \leq C \lambda^n |v| \ \ (v \in E^u_x). $$ $E^{s}$ is the stable subbundle and $E^{u}$ the unstable subbundle. "Continuous splitting" means $x \mapsto E_{x}^{s}$ and $x \mapsto E_{x}^{u}$ are continuous as maps into the Grassmann bundle; the dimensions of $E_{x}^{s}, E_{x}^{u}$ are then locally constant on $Λ$ . A metric in which $C = 1$ may be achieved (an adapted or Lyapunov metric); the constant $C$ measures the failure of the chosen metric to be adapted.

Sign / convention. Throughout, $E^{s}$ is contracted in forward time and $E^{u}$ in backward time, matching Katok-Hasselblatt ^{[Katok-Hasselblatt 1995 §6.1]}; the contraction rate $λ$ and expansion rate $λ^{- 1}$ are uniform over $Λ$ . This is the discrete-time analogue of the Lyapunov-stability sign condition 02.12.08, where a contracting direction certifies stability of an equilibrium.

Definition (Anosov diffeomorphism). $f$ is Anosov if the whole manifold $Λ = M$ is a hyperbolic set. The model is a hyperbolic toral automorphism: $A \in SL (n, Z)$ with no eigenvalue on the unit circle induces $f_{A} : T^{n} \to T^{n}$ , with $E^{s}, E^{u}$ the (constant) sums of generalized eigenspaces for eigenvalues inside and outside the unit circle. An Anosov flow $φ^{t}$ has a splitting $T M = E^{s} \oplus E^{0} \oplus E^{u}$ with $E^{0}$ the one-dimensional flow direction and uniform contraction/expansion on $E^{s} / E^{u}$ ; the geodesic flow on a closed manifold of negative sectional curvature is the canonical example (Anosov 1967 ^{[Anosov 1967]}).

Definition (non-wandering set; Axiom A). Recall from 38.01.01 the non-wandering set $Ω (f) = {x : every neighbourhood U of x has f^{n} (U) \cap U \neq = \emptyset for some n \geq 1}$ . The diffeomorphism $f$ satisfies Axiom A if (i) $Ω (f)$ is a hyperbolic set, and (ii) the periodic points of $f$ are dense in $Ω (f)$ .

Definition (topological transitivity; basic set). A homeomorphism $g$ of a compact set $K$ is topologically transitive if it has a dense orbit, equivalently if for every pair of non-empty open $U, V \subseteq K$ there is $n$ with $g^{n} (U) \cap V \neq = \emptyset$ . A basic set of an Axiom-A system is a closed, $f$ -invariant, topologically transitive subset of $Ω (f)$ that is isolated: it equals $⋂_{n \in Z} f^{n} (V)$ for some neighbourhood $V$ .

Definition (local product structure). A hyperbolic set $Λ$ has local product structure if there is $δ > 0$ such that for $x, y \in Λ$ with $d (x, y) < δ$ the local stable and unstable manifolds meet in a single point of $Λ$ : $W_{ε}^{s} (x) \cap W_{ε}^{u} (y) = {[x, y]} \subseteq Λ$ . Here $W_{ε}^{s} (x) = {y : d (f^{n} x, f^{n} y) \leq ε for all n \geq 0}$ and $W_{ε}^{u}$ is the analogue for $n \leq 0$ ; the stable/unstable manifold theorem identifies these with embedded discs tangent to $E_{x}^{s}, E_{x}^{u}$ .

Counterexamples to common slips

Hyperbolicity is uniform, not pointwise-asymptotic. A splitting in which contraction holds only in the limit (a non-uniform or Pesin hyperbolicity, where $λ$ is replaced by Lyapunov exponents that may degenerate) is not a hyperbolic set in the present sense. The single uniform pair $(C, λ)$ is exactly what yields robustness.
The splitting need not be smooth. $E^{s}, E^{u}$ are continuous and $D f$ -invariant but generally only Hölder, not $C^{1}$ , even for a real-analytic Anosov map. Assuming a smooth splitting is a frequent error; the regularity is a theorem (Hölder), not part of the definition.
Axiom A is more than " $Ω (f)$ hyperbolic". Density of periodic points in $Ω (f)$ is a separate clause. Without it the spectral decomposition can fail; the closing lemma is what makes the two clauses cooperate.
A basic set need not be a manifold. The horseshoe's basic set is a Cantor set, and a general basic set is a compact set with local product structure of Cantor $\times$ disc type, not a submanifold.

Key theorem with proof Intermediate+

Theorem (Smale spectral decomposition). Let $f$ satisfy Axiom A. Then the non-wandering set decomposes as a finite disjoint union $$ \Omega(f) = \Omega_1 \sqcup \Omega_2 \sqcup \cdots \sqcup \Omega_k $$ of closed, $f$ -invariant sets $Ω_{i}$ , each of which is topologically transitive (the basic sets), and the decomposition is unique. Moreover each $Ω_{i}$ further partitions into finitely many disjoint closed sets cyclically permuted by $f$ , on each of which an iterate of $f$ is topologically mixing. (Smale 1967 ^{[Smale 1967]}; Katok-Hasselblatt ^{[Katok-Hasselblatt 1995 §18.3]}.)

Proof. Write $P = Per (f) \cap Ω (f)$ for the periodic points; by Axiom A, $\overline{P} = Ω (f)$ .

A relation on periodic points. For $p, q \in P$ declare $p \sim q$ if $W^{u} (p) ⋔ W^{s} (q) \neq = \emptyset$ and $W^{u} (q) ⋔ W^{s} (p) \neq = \emptyset$ , where $W^{u} (p) = {y : d (f^{- n} y, f^{- n} p) \to 0}$ is the global unstable manifold of the periodic orbit of $p$ , $W^{s} (q)$ the global stable manifold, and $⋔$ denotes transverse intersection. The stable/unstable manifold theorem for the hyperbolic set $Ω (f)$ guarantees these manifolds exist and are injectively immersed discs tangent to $E_{p}^{u}, E_{q}^{s}$ .

The relation is an equivalence relation. Reflexivity holds because $p \in W^{u} (p) \cap W^{s} (p)$ . Symmetry is built into the two-sided definition. Transitivity is the substantive step: if $W^{u} (p)$ meets $W^{s} (q)$ transversally and $W^{u} (q)$ meets $W^{s} (r)$ transversally, the inclination ( $λ$ -) lemma says that forward iterates of a disc in $W^{u} (p)$ accumulate, in the $C^{1}$ topology, on $W^{u} (q)$ ; hence $W^{u} (p)$ also meets $W^{s} (r)$ transversally. The two-sided clause and the symmetric argument give $r \sim p$ , so $\sim$ is transitive.

Finitely many classes, each with closed transitive closure. Each equivalence class is open in $P$ in the following sense: by the stable manifold theorem and transversality, $p \sim q$ is an open condition under moving $p, q$ among periodic points, and the local product structure of $Ω (f)$ (a consequence of hyperbolicity plus the closing lemma) shows the closures of distinct classes are disjoint. Define $Ω_{i} = \overline{[p_{i}]}$ as the closures of the equivalence classes. Compactness of $Ω (f)$ forces finitely many classes. Each $Ω_{i}$ is closed and $f$ -invariant by construction. Topological transitivity of $Ω_{i}$ follows from the Anosov closing lemma: any two points whose orbits come close are shadowed by a genuine periodic orbit, so points of a single class are linked by orbits that pass arbitrarily near any prescribed finite list of targets, producing a dense orbit in $Ω_{i}$ .

Disjointness and exhaustion. Distinct $Ω_{i}$ are disjoint: a point in $Ω_{i} \cap Ω_{j}$ would, by local product structure, force $[p_{i}] \sim [p_{j}]$ . That $⋃_{i} Ω_{i} = Ω (f)$ holds because $\overline{P} = Ω (f)$ and every periodic point lies in some class. Uniqueness: any decomposition into closed invariant transitive pieces must group the periodic points by the same equivalence relation, since transitivity forces all periodic points in a piece to be $\sim$ -related and closedness forces the piece to contain the whole class closure.

Mixing refinement. Fix a basic set $Ω_{i}$ and a periodic point $p \in Ω_{i}$ of period $m$ . The sets $Ω_{i}^{(j)} = \overline{W^{u} (f^{j} p) \cap Ω_{i}}$ , $0 \leq j < m^{'}$ for the appropriate $m^{'} ∣$ (period data), are cyclically permuted by $f$ , pairwise disjoint, and $f^{m^{'}}$ is topologically mixing on each — a consequence of the spectral-gap / local-product structure that promotes transitivity to mixing on the cyclic components. $□$

Bridge. The spectral decomposition builds toward the entire structural-stability and ergodic theory of Axiom-A systems, and the foundational reason it holds is the cooperation of two facts proved from uniform hyperbolicity: the stable/unstable manifold theorem, which gives the transverse intersection geometry, and the closing lemma, which converts near-recurrence into genuine periodic orbits. This is exactly the discrete-time global analogue of the Lyapunov picture 02.12.08: there a single contracting direction at an equilibrium certifies local stability; here a uniform contracting subbundle over all of $Ω (f)$ certifies a global skeleton of basic sets. The decomposition generalises the omega-limit-set and non-wandering-set analysis of 38.01.01 — where $Ω (f)$ was a single closed invariant set — by resolving $Ω (f)$ into irreducible transitive atoms, and it appears again in symbolic dynamics, where each basic set is coded by a subshift of finite type through a Markov partition. The central insight is that hyperbolicity makes recurrence finite-dimensional in a combinatorial sense: putting these together, the qualitative dynamics on $Ω (f)$ reduces to a finite directed graph of basic sets whose internal dynamics is shift-modelled. The bridge is the recognition that the same uniform splitting that drives sensitive dependence also disassembles the recurrent set into a finite list of self-contained mixing machines.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that hyperbolicity of a compact invariant set $Λ$ is independent of the choice of Riemannian metric: if $(C, λ)$ works for a metric $∥ \cdot ∥$ and $∥ \cdot ∥^{'}$ is another smooth metric, then some $(C^{'}, λ)$ with the same $λ$ works for $∥ \cdot ∥^{'}$ after possibly enlarging the constant.

Hint

On a compact set, any two continuous metrics are uniformly equivalent: $a ∥ v ∥ \leq ∥ v ∥^{'} \leq b ∥ v ∥$ for constants $0 < a \leq b$ . Push the equivalence through the contraction estimate.

Answer

By compactness of $Λ$ and continuity, there are constants $0 < a \leq b$ with $a ∥ v ∥ \leq ∥ v ∥^{'} \leq b ∥ v ∥$ for all tangent vectors $v$ over $Λ$ . For $v \in E_{x}^{s}$ , $∥ D f^{n} v ∥^{'} \leq b ∥ D f^{n} v ∥ \leq b C λ^{n} ∥ v ∥ \leq \frac{b C}{a} λ^{n} ∥ v ∥^{'}$ . So the stable estimate holds for $∥ \cdot ∥^{'}$ with $C^{'} = b C / a$ and the same $λ$ ; the unstable estimate is identical with $f^{- 1}$ . The contraction rate $λ$ is metric-independent (it reflects the spectral data of $D f$ ), while the constant $C$ absorbs the metric distortion. This is why one may always pass to an adapted metric with $C = 1$ . Rubric: full credit for the uniform-equivalence step and the same- $λ$ conclusion.

Exercise 4 (medium, symbolic).

Let $Λ$ be a hyperbolic set with adapted metric ( $C = 1$ ). Prove the cone criterion in one direction: if $v = v^{s} + v^{u}$ with $v^{s} \in E_{x}^{s}$ , $v^{u} \in E_{x}^{u}$ and $∥ v^{u} ∥ \geq ∥ v^{s} ∥$ (i.e. $v$ lies in the unstable cone), then $∥ D f_{x} v ∥ \geq λ^{- 1} ∥ v^{u} ∥ - λ ∥ v^{s} ∥$ , and deduce that $D f_{x} v$ again lies in the unstable cone when $λ^{- 2} > 1$ , which always holds.

Hint

Use $D f (v^{s}) \in E_{f (x)}^{s}$ , $D f (v^{u}) \in E_{f (x)}^{u}$ , the adapted bounds $∥ D f v^{s} ∥ \leq λ ∥ v^{s} ∥$ , $∥ D f v^{u} ∥ \geq λ^{- 1} ∥ v^{u} ∥$ , and the orthogonal-ish reverse triangle inequality on the splitting.

Answer

Decompose $D f_{x} v = D f_{x} v^{s} + D f_{x} v^{u}$ with the two summands in $E_{f (x)}^{s}, E_{f (x)}^{u}$ respectively. In the adapted metric, $∥ D f_{x} v^{u} ∥ \geq λ^{- 1} ∥ v^{u} ∥$ and $∥ D f_{x} v^{s} ∥ \leq λ ∥ v^{s} ∥$ . The unstable component of $D f_{x} v$ has norm $\geq λ^{- 1} ∥ v^{u} ∥$ and the stable component has norm $\leq λ ∥ v^{s} ∥$ . Hence the ratio of unstable to stable parts of $D f_{x} v$ is at least $$ \frac{\lambda^{-1}|v^u|}{\lambda|v^s|} = \lambda^{-2}\frac{|v^u|}{|v^s|} \geq \lambda^{-2} \geq \lambda^{-2} > 1 $$ using $∥ v^{u} ∥ \geq ∥ v^{s} ∥$ and $0 < λ < 1$ . So $D f_{x} v$ has a strictly larger unstable-to-stable ratio, hence lies in the (interior of the) unstable cone; the lower bound on $∥ D f_{x} v ∥$ is the reverse triangle inequality $∥ D f_{x} v ∥ \geq ∥ D f_{x} v^{u} ∥ - ∥ D f_{x} v^{s} ∥ \geq λ^{- 1} ∥ v^{u} ∥ - λ ∥ v^{s} ∥$ . This forward-invariance of the unstable cone field is the characterisation that makes hyperbolicity an open condition. Rubric: full credit for the component bounds, the ratio computation, and the cone-invariance conclusion.

Exercise 5 (medium, short-answer).

Prove that a hyperbolic set $Λ$ is expansive: there is $δ_{0} > 0$ such that if $x, y \in Λ$ satisfy $d (f^{n} x, f^{n} y) \leq δ_{0}$ for all $n \in Z$ , then $x = y$ .

Hint

If two orbits stay $δ_{0}$ -close for all time, the separation vector lies in both the local stable manifold (from forward closeness) and the local unstable manifold (from backward closeness). What is $W_{ε}^{s} (x) \cap W_{ε}^{u} (x)$ ?

Answer

Choose $δ_{0} = ε$ smaller than the size on which the local stable/unstable manifold theorem applies. If $d (f^{n} x, f^{n} y) \leq δ_{0}$ for all $n \geq 0$ , then $y \in W_{ε}^{s} (x)$ : forward orbits stay close, which by the contraction on $E^{s}$ and the local product structure means $y$ lies on the local stable manifold of $x$ . Symmetrically, $d (f^{n} x, f^{n} y) \leq δ_{0}$ for all $n \leq 0$ gives $y \in W_{ε}^{u} (x)$ . Therefore $y \in W_{ε}^{s} (x) \cap W_{ε}^{u} (x)$ . These local manifolds are transverse discs tangent to the complementary subspaces $E_{x}^{s}, E_{x}^{u}$ and meet in the single point $x$ for $ε$ small. Hence $y = x$ , and $Λ$ is expansive with constant $δ_{0}$ . Rubric: full credit for placing $y$ in both local manifolds and invoking their transverse single-point intersection.

Exercise 6 (medium, short-answer).

The Smale horseshoe restricted to its invariant Cantor set $Λ$ is conjugate to the full two-sided shift $σ$ on $Σ_{2} = {0, 1}^{Z}$ . Granting this, explain why $Λ$ is a single basic set of an Axiom-A system: verify it is hyperbolic, has dense periodic points, and is topologically transitive.

Hint

Transport the three properties through the conjugacy. The shift has periodic sequences dense and a dense orbit (concatenate all finite words). Hyperbolicity is the horseshoe's defining stretch/fold geometry.

Answer

Hyperbolicity. The horseshoe map stretches the square uniformly along the horizontal direction by a factor $> 1$ and contracts uniformly along the vertical, so on $Λ = ⋂_{n \in Z} f^{n} (Q)$ the tangent bundle splits into horizontal $E^{u}$ and vertical $E^{s}$ with uniform constants — $Λ$ is a hyperbolic set. Dense periodic points. Under the conjugacy $h : Λ \to Σ_{2}$ , periodic points of $f$ correspond to periodic sequences (eventually repeating blocks), which are dense in $Σ_{2}$ in the product topology; pulling back, $Per (f) \cap Λ$ is dense in $Λ$ . Transitivity. The shift has a dense orbit — the sequence concatenating every finite word visits every cylinder — so $σ$ is topologically transitive, and transitivity transfers across the conjugacy to $f ∣_{Λ}$ . The three together (hyperbolic, dense periodic points, transitive) make $Λ$ exactly one basic set; since $Ω (f ∣_{Q}) = Λ$ , the system is Axiom A with a single piece in its spectral decomposition. Rubric: full credit for transporting all three properties and concluding single-basic-set Axiom A.

Exercise 7 (hard, short-answer).

Prove that hyperbolicity is a $C^{1}$ -open condition on the set $Λ$ in the following sense: if $Λ$ is a hyperbolic set for $f$ with the adapted-metric cone field, then there is a $C^{1}$ -neighbourhood $U$ of $f$ and a neighbourhood $V \supseteq Λ$ such that for every $g \in U$ , the maximal $g$ -invariant set in $V$ is hyperbolic. (Sketch the cone-field argument; you may cite the stable manifold theorem.)

Hint

The cone criterion (Exercise 4) is an open inequality on $D g$ . Forward-invariance of the unstable cone field and backward-invariance of the stable cone field persist under $C^{1}$ -small perturbation, and invariant cone fields with strict expansion/contraction reconstruct a hyperbolic splitting by intersecting iterated cones.

Answer

Equip $Λ$ with the adapted metric and the unstable cone field $C_{x}^{u} = {v^{s} + v^{u} : ∥ v^{u} ∥ \geq ∥ v^{s} ∥}$ and stable cone field $C_{x}^{s}$ . By Exercise 4, $D f_{x} (C_{x}^{u}) \subseteq int C_{f (x)}^{u}$ with strict expansion of the unstable part, and dually $D f_{x}^{- 1} (C_{x}^{s}) \subseteq int C_{f^{- 1} (x)}^{s}$ . These are strict open inclusions and strict norm inequalities. Extend the cone fields continuously to a neighbourhood $V$ of $Λ$ . The maps $g \mapsto D g$ are continuous in the $C^{1}$ topology, so for $g$ in a small $C^{1}$ -neighbourhood $U$ of $f$ and points in $V$ , the perturbed derivatives $D g$ still satisfy $D g (C^{u}) \subseteq int C^{u}$ and $D g^{- 1} (C^{s}) \subseteq int C^{s}$ with expansion/contraction factors close to $λ^{\mp 1}$ , hence still $> 1$ and $< 1$ . Let $Λ_{g} = ⋂_{n} g^{n} (V)$ be the maximal invariant set. For $x \in Λ_{g}$ define $E_{x}^{u} = ⋂_{n \geq 0} D g^{n} (C_{g^{- n} x}^{u})$ and $E_{x}^{s} = ⋂_{n \geq 0} D g^{- n} (C_{g^{n} x}^{s})$ ; the nested intersections of forward-invariant cones with uniform expansion collapse to subspaces of complementary dimension, giving a continuous $D g$ -invariant splitting with uniform constants inherited from the cone factors. Thus $Λ_{g}$ is hyperbolic for $g$ . Rubric: full credit for strict cone-invariance, persistence under $C^{1}$ -perturbation, and the cone-intersection reconstruction of the splitting.

Exercise 8 (hard, short-answer).

State the Anosov closing lemma and use it to prove that for an Axiom-A system, periodic points are dense in $Ω (f)$ follows from the shadowing property — i.e., explain the logical role the closing lemma plays in making the spectral decomposition's transitivity argument work.

Hint

Closing lemma: an orbit segment that nearly returns is uniformly shadowed by a genuine periodic orbit. Apply it to a non-wandering point, whose neighbourhood returns to meet itself.

Answer

Anosov closing lemma. There exist $ε_{0} > 0$ and $L > 0$ such that if $x$ in a hyperbolic set satisfies $d (f^{N} x, x) < ε$ for some $N$ and $ε < ε_{0}$ , then there is a genuine periodic point $p$ with $f^{N} p = p$ and $d (f^{n} x, f^{n} p) \leq L ε$ for $0 \leq n \leq N$ . Role in density of periodic points. Let $x \in Ω (f)$ and $U$ a neighbourhood. Non-wandering means some iterate returns: there is $y$ near $x$ and $N$ with $f^{N} y$ near $y$ , so $d (f^{N} y, y) < ε$ . The closing lemma produces a periodic $p$ shadowing the segment to within $L ε$ , hence $p \in U$ for $ε$ small. So periodic points are dense in $Ω (f)$ — this is precisely Axiom A clause (ii), recovered from hyperbolicity plus near-recurrence. Role in transitivity. In the spectral decomposition proof, transitivity of a basic set requires building an orbit that passes near any prescribed finite list of points in the class. The closing lemma (and its relative, the shadowing lemma, which strings together pseudo-orbits) converts a concatenation of orbit segments linking those points into a genuine orbit that shadows the whole pseudo-orbit, producing the required dense orbit. Without shadowing, the equivalence-class structure on periodic points would not assemble into closed transitive sets. Rubric: full credit for the correct statement, the density derivation, and articulating the transitivity role.

Advanced results Master

Theorem (stable and unstable manifold theorem for hyperbolic sets). Let $Λ$ be a hyperbolic set for a $C^{r}$ diffeomorphism $f$ ( $r \geq 1$ ). For small $ε > 0$ and each $x \in Λ$ , the local stable set $W_{ε}^{s} (x) = {y : d (f^{n} x, f^{n} y) \leq ε, n \geq 0}$ is a $C^{r}$ embedded disc tangent to $E_{x}^{s}$ at $x$ , varying continuously with $x$ ; likewise $W_{ε}^{u} (x)$ tangent to $E_{x}^{u}$ . 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. (Katok-Hasselblatt ^{[Katok-Hasselblatt 1995 §6.2]}; Hirsch-Pugh-Shub graph transform.)

The manifolds are constructed as the unique fixed point of the graph transform, a contraction on a Banach space of sections of the unstable cone bundle: a graph over $E^{u}$ is pushed forward and the uniform expansion makes the induced operator a contraction in the $C^{0}$ (then $C^{r}$ ) section norm, so a unique invariant graph exists and is the local unstable manifold. The tangency to $E_{x}^{u}$ and the continuous (Hölder) dependence on $x$ are read off the fixed-point estimates. This is the analytic engine beneath every structural result in the subject, and it is 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.

Theorem (structural stability of Anosov diffeomorphisms). Every Anosov diffeomorphism $f$ is structurally stable: there is a $C^{1}$ -neighbourhood $U$ of $f$ such that every $g \in U$ is topologically conjugate to $f$ — there is a homeomorphism $h$ of $M$ with $h \circ f = g \circ h$ , and $h$ is $C^{0}$ -close to the identity. (Anosov 1967 ^{[Anosov 1967]}.)

The conjugacy is produced by structural-stability via shadowing: a $g$ -orbit is a pseudo-orbit for $f$ , and the shadowing lemma assigns it a unique nearby genuine $f$ -orbit; the assignment $h$ is a homeomorphism intertwining the two. Anosov diffeomorphisms are therefore $C^{1}$ -robust as dynamical systems — their entire conjugacy class is open — which is the structural-stability conclusion Smale's programme sought, realised at the level of the whole manifold rather than a single basic set.

Theorem ( $Ω$ -stability and the no-cycle condition). An Axiom-A diffeomorphism whose basic sets satisfy the no-cycle condition — there is no cyclic chain $Ω_{i_{0}} \to Ω_{i_{1}} \to \dots \to Ω_{i_{0}}$ of distinct basic sets linked by heteroclinic orbits — is $Ω$ -stable: every $C^{1}$ -nearby $g$ has its non-wandering set $Ω (g)$ topologically conjugate to $Ω (f)$ via the restriction of a homeomorphism. (Smale 1970; Palis 1970.)

The basic sets form the vertices of a directed graph (the phase diagram), with an edge $Ω_{i} \to Ω_{j}$ when $W^{u} (Ω_{i}) \cap W^{s} (Ω_{j}) \neq = \emptyset$ . The no-cycle condition is acyclicity of this graph, and it is exactly the hypothesis that promotes $Ω$ -stability to structural stability of the whole system. The cat map and any transitive Anosov map have a single basic set ( $Ω (f) = M$ ), acyclic for the plain reason that one vertex admits no cycle; the horseshoe-with-attached-sink models show how cycles obstruct stability.

Theorem (Markov partitions and symbolic coding). Every basic set $Ω_{i}$ of an Axiom-A system admits a Markov partition into rectangles with local product structure, inducing a finite-to-one semiconjugacy from a subshift of finite type $σ_{A}$ onto $f ∣_{Ω_{i}}$ ; the coding is a conjugacy off a small exceptional set. (Bowen 1975 ^{[Bowen 1975]}; Sinai for Anosov.)

A Markov partition ${R_{1}, \dots, R_{m}}$ has the property that $f (W^{u} (x) \cap R_{i}) \supseteq W^{u} (f x) \cap R_{j}$ whenever the transition $i \to j$ is allowed, which is precisely the Markov compatibility making the itinerary map a subshift-of-finite-type coding with transition matrix $A_{ij} \in {0, 1}$ . The symbolic model reduces the ergodic theory of $f ∣_{Ω_{i}}$ — topological entropy $lo g$ (spectral radius of $A$ ), the variational principle, equilibrium states, and the SRB measure — to the combinatorics of $σ_{A}$ , the construction Bowen used to build the thermodynamic formalism for Anosov and Axiom-A systems.

Synthesis. The four theorems are one structural story: uniform hyperbolicity, through the graph transform, manufactures the invariant manifolds; through shadowing it manufactures the conjugacies; and through Markov partitions it manufactures the symbolic codes. The central insight is that the single quantitative datum — a uniform splitting $E^{s} \oplus E^{u}$ with constants $(C, λ)$ — is exactly what is needed to make the recurrent dynamics simultaneously rigid (structurally and $Ω$ -stable) and computable (coded by a subshift of finite type), and this is the foundational reason Smale's programme succeeds for hyperbolic systems where it fails in general. This is exactly the global-over- $Ω (f)$ generalisation of the local Lyapunov-linearization theorem 02.12.08: a Hurwitz spectrum at one equilibrium gives a local stable manifold and local structural stability, while a uniform hyperbolic splitting over the whole non-wandering set gives global stable foliations and global structural stability. The spectral decomposition is dual to the symbolic coding — the decomposition resolves $Ω (f)$ into transitive atoms and the coding resolves each atom into a subshift — and putting these together yields the working picture of hyperbolic dynamics: a finite acyclic graph of basic sets, each a topologically mixing subshift of finite type, robust under perturbation. The bridge is that hyperbolicity converts the analytic problem of long-term behaviour into the finite combinatorial problem of a transition matrix, generalising the orbit/limit-set/conjugacy framework of 38.01.01 to its sharpest and most structured form.

Full proof set Master

Proposition (uniqueness and continuity of the hyperbolic splitting). On a hyperbolic set $Λ$ , the splitting $E^{s} \oplus E^{u}$ is unique and the subbundles depend continuously on the point.

Proof. Uniqueness. Suppose $T_{Λ} M = E^{s} \oplus E^{u} = \tilde{E}^{s} \oplus \tilde{E}^{u}$ are two hyperbolic splittings with the same constants (any two admissible splittings can be compared after passing to a common metric). A vector $v \in \tilde{E}_{x}^{s}$ has $∥ D f^{n} v ∥ \leq C λ^{n} ∥ v ∥ \to 0$ . Write $v = v^{s} + v^{u}$ in the first splitting; then $∥ D f^{n} v^{u} ∥ = ∥ D f^{n} v - D f^{n} v^{s} ∥$ , and since $∥ D f^{n} v ∥ \to 0$ while $∥ D f^{n} v^{s} ∥ \to 0$ , also $∥ D f^{n} v^{u} ∥ \to 0$ . But $v^{u} \in E_{x}^{u}$ satisfies $∥ D f^{n} v^{u} ∥ \geq C^{- 1} λ^{- n} ∥ v^{u} ∥ \to \infty$ unless $v^{u} = 0$ . Hence $v^{u} = 0$ and $\tilde{E}_{x}^{s} \subseteq E_{x}^{s}$ ; the reverse inclusion is symmetric, so $\tilde{E}^{s} = E^{s}$ , and dually $\tilde{E}^{u} = E^{u}$ . Continuity. Let $x_{k} \to x$ in $Λ$ and pass to a subsequence so that $E_{x_{k}}^{s}$ converges in the Grassmannian to some subspace $F$ . For $v \in F$ , take $v_{k} \in E_{x_{k}}^{s}$ with $v_{k} \to v$ ; then $∥ D f^{n} v ∥ = lim_{k} ∥ D f^{n} v_{k} ∥ \leq C λ^{n} ∥ v ∥$ for every $n$ , so $v \in E_{x}^{s}$ by the contraction characterisation. Thus $F \subseteq E_{x}^{s}$ , and equality holds by matching dimensions (which are locally constant). Every convergent subsequence has limit $E_{x}^{s}$ , so $E_{x_{k}}^{s} \to E_{x}^{s}$ ; the unstable case is identical. $□$

Proposition (the dynamical characterisation of $E^{s}$ ). For $x$ in a hyperbolic set, $E_{x}^{s} = {v \in T_{x} M : sup_{n \geq 0} ∥ D f_{x}^{n} v ∥ < \infty}$ , and this set is exactly the vectors with $∥ D f_{x}^{n} v ∥ \to 0$ .

Proof. If $v \in E_{x}^{s}$ then $∥ D f^{n} v ∥ \leq C λ^{n} ∥ v ∥ \to 0$ , so $v$ is in the right-hand set and the supremum is finite. Conversely, let $sup_{n} ∥ D f^{n} v ∥ =: B < \infty$ and write $v = v^{s} + v^{u}$ . Then $D f^{n} v^{u} = D f^{n} v - D f^{n} v^{s}$ , so $∥ D f^{n} v^{u} ∥ \leq B + C λ^{n} ∥ v^{s} ∥ \leq B + C ∥ v^{s} ∥$ is bounded; but $∥ D f^{n} v^{u} ∥ \geq C^{- 1} λ^{- n} ∥ v^{u} ∥$ grows without bound unless $v^{u} = 0$ . Hence $v = v^{s} \in E_{x}^{s}$ . The boundedness condition and the decay condition therefore define the same set, namely $E_{x}^{s}$ . $□$

Proposition (finiteness of the spectral decomposition). The number of basic sets in the Smale decomposition is finite.

Proof. Each basic set $Ω_{i}$ is the closure of an equivalence class of periodic points under the relation $\sim$ of the Key theorem, and is an isolated set: $Ω_{i} = ⋂_{n} f^{n} (V_{i})$ for an open $V_{i} \supseteq Ω_{i}$ with $\overline{V_{i}}$ disjoint from the other basic sets, by local product structure. The sets $V_{i}$ are pairwise disjoint open subsets each meeting the compact set $Ω (f)$ . A compact set cannot contain an infinite family of pairwise disjoint open sets each of which meets it in a relatively open non-empty piece while the closures stay separated: choosing one point $p_{i} \in Ω_{i}$ from each, the sequence $(p_{i})$ would have a convergent subsequence with limit $p \in Ω (f)$ , and $p$ lies in some basic set $Ω_{j}$ whose neighbourhood $V_{j}$ then contains infinitely many $p_{i}$ , contradicting the disjointness of the $V_{i}$ and the separation of distinct classes. Hence only finitely many basic sets occur. $□$

Proposition (entropy of the cat map equals $lo g λ$ ). For the hyperbolic toral automorphism $f_{A}$ , $A \in SL (2, Z)$ with eigenvalues $λ > 1 > λ^{- 1}$ , the topological entropy is $h_{top} (f_{A}) = lo g λ$ .

Proof. The unstable subbundle $E^{u}$ is the constant line of slope the expanding eigendirection, and $D f_{A} = A$ expands it by exactly $λ$ . By the volume-growth / Yomdin-Newhouse principle for $C^{\infty}$ maps — or directly via the periodic-point count $# Fix (f_{A}^{n}) = ∣ det (A^{n} - I) ∣ = ∣ λ^{n} + λ^{- n} - 2∣$ established for toral automorphisms — the exponential growth rate of periodic orbits is $lim_{n} \frac{1}{n} lo g # Fix (f_{A}^{n}) = lo g λ$ . For Axiom-A (here Anosov) systems the topological entropy equals this periodic-orbit growth rate (Bowen's equidistribution of periodic orbits, a consequence of the Markov coding), so $h_{top} (f_{A}) = lo g λ$ . Equivalently, the Markov partition codes $f_{A}$ by a subshift of finite type whose transition matrix has spectral radius $λ$ , and the entropy of a subshift of finite type is the log of that spectral radius. $□$

Proposition (basic sets are isolated, hence the decomposition is into isolated invariant sets). Each $Ω_{i}$ equals $⋂_{n \in Z} f^{n} (V)$ for some open $V \supseteq Ω_{i}$ .

Proof. Local product structure provides, for each $x \in Ω_{i}$ , a product neighbourhood $[W_{ε}^{u} (x) \cap Ω_{i}] \times [W_{ε}^{s} (x) \cap Ω_{i}]$ within $Ω_{i}$ , and expansivity (Exercise 5) gives an expansivity constant $δ_{0}$ . Take $V = {y : d (y, Ω_{i}) < δ_{0} /2}$ . If $y \in ⋂_{n} f^{n} (V)$ , then $d (f^{n} y, Ω_{i}) < δ_{0} /2$ for all $n$ ; choosing $x_{n} \in Ω_{i}$ with $d (f^{n} y, x_{n})$ small and using compactness plus the closing/shadowing lemma, the orbit of $y$ is shadowed by an orbit in $Ω_{i}$ within $δ_{0}$ , and expansivity forces $y$ to coincide with that shadowing orbit's base point, so $y \in Ω_{i}$ . The reverse inclusion $Ω_{i} \subseteq ⋂_{n} f^{n} (V)$ is immediate from invariance. Hence $Ω_{i}$ is isolated. $□$

Connections Master

Dynamical systems, orbits, and limit sets 38.01.01. This unit specialises the non-wandering set $Ω (f)$ defined there to the hyperbolic case and resolves it into basic sets. The conjugacy invariants developed in the foundational unit — orbits, periodic points, $Ω (f)$ , topological entropy — are exactly what the spectral decomposition organises, and the canonical examples (cat map, horseshoe) introduced there as conjugate-to-shift systems are reinterpreted here as the model Anosov diffeomorphism and the model hyperbolic basic set. The earlier unit supplies the topological-dynamics vocabulary; this unit supplies the differentiable structure that makes recurrence finite and robust.
Phase flow / one-parameter group 02.12.02. Anosov flows — the continuous-time analogue treated here — are one-parameter groups $φ^{t}$ with a hyperbolic splitting $T M = E^{s} \oplus E^{0} \oplus E^{u}$ along the flow direction $E^{0}$ . The geodesic flow on a negatively curved manifold is the motivating example, and the flow-box and one-parameter-group structure imported from 02.12.02 is the substrate on which the continuous-time hyperbolicity estimates are stated. The discrete-time spectral decomposition has a flow analogue decomposing the non-wandering set into transitive hyperbolic basic pieces.
Lyapunov stability (direct method) 02.12.08. Hyperbolicity is the global, uniform-over-an-invariant-set sharpening of the local Lyapunov-linearization picture: a single hyperbolic equilibrium with a Hurwitz stable subspace and an anti-Hurwitz unstable subspace is the one-point hyperbolic set, its local stable manifold is the $W^{s}$ of this unit's manifold theorem, and the quadratic Lyapunov function constructed from the Lyapunov equation is the adapted-metric norm that realises $C = 1$ . The contraction estimate $∥ D f^{n} v ∥ \leq C λ^{n} ∥ v ∥$ on $E^{s}$ is the discrete-time, uniform-constant version of the exponential decay $∥ e^{A t} ∥ \leq C e^{- α t}$ for a Hurwitz linearization.
Symbolic dynamics / subshifts of finite type 38.02.01. Markov partitions code each basic set by a subshift of finite type, transporting topological entropy, the variational principle, and equilibrium states to the combinatorics of a transition matrix. This is the bridge by which the differentiable theory of hyperbolic sets becomes the combinatorial theory of shift spaces, and it is the technical heart of Bowen's thermodynamic formalism.

Historical & philosophical context Master

Uniform hyperbolicity emerged from two independent sources in the 1960s. Dmitri Anosov, in his 1967 Steklov Institute monograph Geodesic flows on closed Riemannian manifolds of negative curvature ^{[Anosov 1967]}, isolated the uniform contraction/expansion condition — now called the Anosov condition — and proved that geodesic flows on negatively curved manifolds satisfy it and are structurally stable, settling a robustness question going back to Hadamard's and Hedlund's study of geodesics on surfaces of negative curvature. Stephen Smale, in the same period and culminating in his 1967 Bulletin of the American Mathematical Society survey Differentiable dynamical systems ^{[Smale 1967]}, introduced the horseshoe as a structurally stable source of infinitely many periodic orbits, formulated Axiom A, and proved the spectral decomposition of the non-wandering set into basic sets — the result that turned a zoo of examples into a structural theory.

The synthesis was carried forward by Rufus Bowen, whose 1975 Springer Lecture Notes Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms ^{[Bowen 1975]} built Markov partitions and subshift-of-finite-type codings on every basic set, importing Sinai's and Ruelle's thermodynamic formalism and constructing the measures now bearing the Sinai-Ruelle-Bowen name. The structural-stability conjecture — that structurally stable systems are exactly those satisfying Axiom A plus strong transversality — was proved by Mañé (1988) in the $C^{1}$ category, completing the programme Smale articulated. Katok and Hasselblatt's 1995 Introduction to the Modern Theory of Dynamical Systems ^{[Katok-Hasselblatt 1995]} is the canonical modern reference, and the subsequent non-uniform theory of Pesin extended the hyperbolic paradigm to systems hyperbolic only on a set of full measure.

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{Smale1967,
  author  = {Smale, Stephen},
  title   = {Differentiable dynamical systems},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {73},
  year    = {1967},
  pages   = {747--817}
}

@article{Anosov1967,
  author  = {Anosov, Dmitri V.},
  title   = {Geodesic flows on closed {R}iemannian manifolds of negative curvature},
  journal = {Proceedings of the Steklov Institute of Mathematics},
  volume  = {90},
  year    = {1967}
}

@book{Bowen1975,
  author    = {Bowen, Rufus},
  title     = {Equilibrium States and the Ergodic Theory of {A}nosov Diffeomorphisms},
  publisher = {Springer-Verlag},
  series    = {Lecture Notes in Mathematics},
  volume    = {470},
  year      = {1975}
}

@book{BrinStuck2002,
  author    = {Brin, Michael and Stuck, Garrett},
  title     = {Introduction to Dynamical Systems},
  publisher = {Cambridge University Press},
  year      = {2002}
}

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

@article{Mane1988,
  author  = {Ma{\~n}{\'e}, Ricardo},
  title   = {A proof of the {$C^1$} stability conjecture},
  journal = {Publications Math\'ematiques de l'IH\'ES},
  volume  = {66},
  year    = {1988},
  pages   = {161--210}
}

@book{HasselblattKatok2003,
  author    = {Hasselblatt, Boris and Katok, Anatole},
  title     = {A First Course in Dynamics},
  publisher = {Cambridge University Press},
  year      = {2003}
}

Prerequisites

38.01.01
02.12.02
02.12.08

Tier anchors

beginner: Strogatz 1994 *Nonlinear Dynamics and Chaos* (Addison-Wesley) Ch. 10–12 (stretching-and-folding, the cat map and saddle pictures); 3Blue1Brown *Chaos* and the Arnold-cat-map visual essays for the stable/unstable-direction imagery
intermediate: Brin-Stuck 2002 *Introduction to Dynamical Systems* (Cambridge University Press) Ch. 5 §5.1–§5.4 (hyperbolic sets, Anosov diffeomorphisms, the stable manifold theorem, structural stability); Robinson 1999 *Dynamical Systems* (CRC) Ch. 5
master: Katok-Hasselblatt 1995 *Introduction to the Modern Theory of Dynamical Systems* (Cambridge) Ch. 6 (hyperbolic sets), Ch. 17–18 (Anosov systems, Axiom A, the spectral decomposition); Shub 1987 *Global Stability of Dynamical Systems* (Springer); Smale 1967 *Differentiable dynamical systems* (Bull. Amer. Math. Soc. 73) — the originating survey of Axiom A, basic sets, and the spectral decomposition; Bowen 1975 *Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms* (Springer LNM 470)

References

Katok, Hasselblatt 1995 — Introduction to the Modern Theory of Dynamical Systems · Ch. 6 §6.1–§6.5 (hyperbolic sets, the invariant splitting, cone criterion, expansiveness, the shadowing lemma), Ch. 17 (Anosov diffeomorphisms and flows, structural stability), Ch. 18 §18.3 (the Smale spectral decomposition of an Axiom-A non-wandering set into basic sets, the spectral and decomposition theorems); Encyclopedia of Mathematics and its Applications 54, Cambridge University Press 1995
Smale 1967 — Differentiable dynamical systems · Bulletin of the American Mathematical Society 73 (1967) 747–817 §I.5–§II (Axiom A, the spectral decomposition theorem $\Omega(f) = \Omega_1 \sqcup \cdots \sqcup \Omega_k$ into closed transitive basic sets, the no-cycle condition, $\Omega$-stability, and the horseshoe as the model hyperbolic basic set)
Brin, Stuck 2002 — Introduction to Dynamical Systems · Ch. 5 §5.1–§5.4 (hyperbolic sets and the splitting $E^s \oplus E^u$, Anosov diffeomorphisms, hyperbolic toral automorphisms, the stable/unstable manifold theorem, local product structure, structural stability); Cambridge University Press 2002
Anosov 1967 — Geodesic flows on closed Riemannian manifolds of negative curvature · Proceedings of the Steklov Institute of Mathematics 90 (1967); originator of the uniform-hyperbolicity (Anosov) condition and the structural stability of geodesic flows on negatively curved manifolds and of Anosov diffeomorphisms
Bowen 1975 — Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms · Springer Lecture Notes in Mathematics 470; the shadowing lemma, Markov partitions of basic sets, symbolic coding by subshifts of finite type, and the ergodic theory built on the spectral decomposition
Hasselblatt, Katok 2003 — A First Course in Dynamics · Cambridge University Press 2003, Ch. 7 (the linear and toral models of hyperbolicity), Ch. 10 (hyperbolic dynamics, the horseshoe and the cat map as model hyperbolic sets)

Estimated time

beginner: 19m
intermediate: 44m
master: 86m