Dynamical Systems: Hyperbolic Dynamics¶

Job: develop the core of modern smooth dynamics — hyperbolic sets and Anosov diffeomorphisms (Smale–Anosov–Bowen), the Hadamard–Perron stable/unstable manifolds, the shadowing and closing lemmas, Markov partitions and symbolic dynamics, and Smale's Axiom A / $\Omega$-stability theorem with the spectral decomposition.

Core¶

Hyperbolic sets and Anosov diffeomorphisms¶

Let $f:M\to M$ be a $C^r$ diffeomorphism of a compact Riemannian manifold. A compact invariant set $\Lambda=f(\Lambda)$ is hyperbolic if the tangent bundle over $\Lambda$ splits as a continuous $Df$-invariant Whitney sum

\[ T_\Lambda M = E^s\oplus E^u,\qquad \|Df^n|_{E^s(x)}\|\le C\lambda^n,\;\;\|Df^{-n}|_{E^u(x)}\|\le C\lambda^n \]

for all $x\in\Lambda$, $n\ge 0$, some $C\ge 1$, $\lambda<1$. $E^s$ is the stable bundle (uniformly contracted forward), $E^u$ the unstable (contracted backward). An Anosov diffeomorphism is one where $\Lambda=M$ (the entire manifold is hyperbolic); an Anosov flow is defined analogously (the splitting $E^{ss}\oplus E^0\oplus E^{uu}$ with a 1-dim flow direction $E^0$).

Theorem (Anosov 1967; structural stability). Anosov diffeomorphisms are structurally stable in the $C^1$ topology: for any $C^1$-small perturbation $g$ of an Anosov $f$, there is a homeomorphism $h$ with $h\circ f = g\circ h$ (topological conjugacy). Anosov flows are similarly structurally stable.

Examples: the Arnold cat map $f:(x,y)\mapsto(2x+y,x+y)$ on $\mathbb{T}^2$ (Anosov); geodesic flows on compact negatively curved manifolds (Anosov flows, Anosov 1967); the Smale–Williams solenoid (a hyperbolic attractor, not Anosov).

Stable/unstable manifolds (Hadamard–Perron)¶

Theorem (Hadamard–Perron). For $f$ a $C^r$ diffeomorphism and $\Lambda$ hyperbolic, each $x\in\Lambda$ has local stable and unstable manifolds $W^{s}_{\mathrm{loc}}(x),W^{u}_{\mathrm{loc}}(x)\subset M$, $C^r$ embedded disks tangent to $E^s(x),E^u(x)$ at $x$, with $f(W^{s}_{\mathrm{loc}}(x))\subset W^{s}_{\mathrm{loc}}(f(x))$, $f^{-1}(W^{u}_{\mathrm{loc}}(x))\subset W^{u}_{\mathrm{loc}}(f^{-1}(x))$, and the global manifolds $W^{s/u}(x)=\bigcup_n f^{\mp n}(W^{s/u}_{\mathrm{loc}}(f^n x))$. The stable manifold $W^s(x)=\{y\mid d(f^n x,f^n y)\to 0\}$, the unstable $W^u(x)=\{y\mid d(f^{-n} x,f^{-n} y)\to 0\}$.

The Hadamard–Perron theorem is the geometric backbone: hyperbolicity gives smooth invariant manifolds tangent to the stable/unstable bundles, and the global $W^{s/u}$ are the sets of points asymptotic to $x$ forward/backward. The $C^r$ regularity of $W^{s/u}$ matches $f$ (Hadamard–Perron for $r<\infty$; for $r=\infty$ the manifolds are $C^\infty$ under standard hypotheses).

Shadowing and closing lemmas¶

Theorem (shadowing lemma). For $f$ with hyperbolic set $\Lambda$, for every $\varepsilon>0$ there is $\delta>0$ such that every $\delta$-pseudo-orbit $\{x_n\}$ in $\Lambda$ (i.e. $d(f(x_n),x_{n+1})<\delta$) is $\varepsilon$-shadowed by a true orbit: there exists $y$ with $d(f^n y,x_n)<\varepsilon$ for all $n$.

Theorem (Anosov closing lemma). For $f$ Anosov, for every $\varepsilon>0$ there is $\delta>0$ such that any $\delta$-pseudo-periodic orbit $\{x_0,\dots,x_{p-1}\}$ (with $d(f(x_{p-1}),x_0)<\delta$) is $\varepsilon$-shadowed by a periodic orbit: there exists a periodic point $y$ of period $p$ with $d(f^n y,x_n)<\varepsilon$. The periodic orbits are dense in the non-wandering set of an Anosov diffeomorphism.

The shadowing / closing lemmas are the engine of hyperbolic dynamics: pseudo-orbits (which are easy to construct, e.g. via symbolic dynamics) are shadowed by true orbits, and pseudo-periodic orbits are shadowed by genuine periodic orbits — giving density of periodic points and the structural / spectral results.

Markov partitions and symbolic dynamics¶

Theorem (Anosov–Bowen, Markov partitions for Axiom A). For an Axiom A diffeomorphism (see below) with a basic set $\Lambda$, there exists a Markov partition $\mathcal{R}=\{R_1,\dots,R_N\}$ of $\Lambda$ into "rectangles" with the Markov property (the image of $W^u(R_i)\cap R_j$ under $f$ crosses $R_j$ in the unstable direction), and a subshift of finite type $\Sigma_A\subset\{1,\dots,N\}^{\mathbb{Z}$ (the two-sided shift with transition matrix $A_{ij}=1$ iff $f(\mathrm{int}R_i)\cap\mathrm{int}R_j\ne\emptyset$) with a semi-conjugacy $\pi:\Sigma_A\to\Lambda$, $\pi\circ\sigma=f\circ\pi$, finite-to-one and one-to-one off a meager set.

The Markov partition encodes the smooth dynamics as a subshift of finite type (a purely combinatorial / symbolic object); the symbolic coding is the bridge from smooth dynamics to ergodic theory (via invariant measures on $\Sigma_A$) and to topological entropy.

Hyperbolic dynamics notion	Symbolic dynamics counterpart
Hyperbolic set $\Lambda$	Subshift of finite type $\Sigma_A$
$f	_\Lambda$
Periodic orbit of $f$	Periodic sequence in $\Sigma_A$
Topological entropy $h_{\mathrm{top}}(f	_\Lambda)$
Markov partition	The alphabet $\{1,\dots,N\}$ + transition matrix $A$
Smale horseshoe	Full 2-shift $\{0,1\}^{\mathbb{Z}$
Bowen measure (max entropy)	Parry measure on $\Sigma_A$ (Markov measure of max entropy)
Stable/unstable manifolds	Stable/unstable cylinders in $\Sigma_A$

Axiom A and the $\Omega$-stability theorem (Smale)¶

Definition (Smale's Axiom A). A diffeomorphism $f$ satisfies Axiom A if (i) the non-wandering set $\Omega(f)$ is hyperbolic, and (ii) the periodic points are dense in $\Omega(f)$.

Theorem (Smale spectral decomposition). For Axiom A $f$, the non-wandering set decomposes as a finite disjoint union of basic sets $\Omega(f)=\Lambda_1\cup\cdots\cup\Lambda_N$, each $\Lambda_i$ compact invariant hyperbolic, topologically transitive (with a dense orbit), and $f|_{\Lambda_i}$ is the spectral component. The decomposition is unique.

Theorem (Smale $\Omega$-stability, 1970). An Axiom A diffeomorphism with the no-cycle condition (no cyclic ordering among the basic sets) is $\Omega$-stable: the non-wandering set of any $C^1$-small perturbation is topologically conjugate to $\Omega(f)$. The no-cycle condition is the sharp obstruction — cycles among basic sets break $\Omega$-stability.

Axiom A + no-cycles is the cleanest structural-stability statement beyond the Anosov case; it captures the hyperbolic dynamics whose non-wandering set is a finite union of basic pieces, each symbolically coded by a Markov partition.

See it¶

The Arnold cat map $f:(x,y)\mapsto(2x+y,x+y)$ on $\mathbb{T}^2$ — hyperbolic splitting, Markov partition, topological entropy. The linear map $A=\begin{pmatrix}2&1\\1&1\end{pmatrix}$ has eigenvalues $\lambda_+=\frac{3+\sqrt 5}{2}>1$ and $\lambda_-=\frac{3-\sqrt 5}{2}<1$, with eigendirections $v_+=(\frac{1+\sqrt 5}{2},1)$ (the golden-ratio slope) and $v_-=(\frac{1-\sqrt 5}{2},1)$. The hyperbolic splitting is $T\mathbb{T}^2=E^s\oplus E^u$ with $E^s=\mathbb{R}v_-$ (contracted by $\lambda_-$), $E^u=\mathbb{R}v_+$ (expanded by $\lambda_+$) — the splitting is constant (the map is linear), and $\mathbb{T}^2$ is Anosov.

The Markov partition: the cat map admits a Markov partition into two rectangles (the standard two-rectangle partition based on the eigenlines mod 1), with transition matrix $A=\begin{pmatrix}1&1\\1&0\end{pmatrix}$ (or a permutation thereof depending on the rectangle labeling). The Markov partition gives a semi-conjugacy $\pi:\Sigma_A\to\mathbb{T}^2$ where $\Sigma_A$ is the subshift of finite type with transition matrix $A$ (the Fibonacci subshift, since $\det A=-1$ and the number of allowed words of length $n$ grows like the Fibonacci numbers).

The topological entropy: $h_{\mathrm{top}}(f)=\log\rho(A)=\log\frac{3+\sqrt 5}{2}=\log\lambda_+$, the log of the spectral radius of the transition matrix, which equals the log of the expanding eigenvalue of the linear map (the entropy of a linear toral automorphism is the sum of the positive log-Lyapunov exponents, here a single exponent). The Markov coding identifies the smooth dynamics with the Fibonacci subshift, and the entropy is read off the transition matrix — the operational content of "hyperbolic dynamics reduces to symbolic dynamics, and the entropy is the log of the spectral radius of the transition matrix."

Check¶

Prove that the periodic points of the Arnold cat map are dense in $\mathbb{T}^2$, using the closing lemma and the rational-torus structure.

The cat map $A\in\mathrm{SL}(2,\mathbb{Z})$ has $A^k-I$ invertible over $\mathbb{Q}$ for infinitely many $k$ (no eigenvalue of $A$ is a root of unity, since $\lambda_\pm$ are real $\ne\pm 1$). For such $k$, $\mathrm{Fix}(A^k)=\ker(A^k-I)\cap\mathbb{T}^2$ is a finite subgroup of $\mathbb{T}^2$ of order $|\det(A^k-I)|\to\infty$ as $k\to\infty$, and these fixed points are uniformly distributed in $\mathbb{T}^2$ (Weyl / equidistribution of rational points with growing denominators). Hence the periodic points are dense. The closing-lemma route: for any $x\in\mathbb{T}^2$ and $\varepsilon>0$, take a long pseudo-periodic orbit near $x$ (possible by the hyperbolicity and the density of rational returns); the Anosov closing lemma gives a genuine periodic point $\varepsilon$-close to $x$, so the periodic points are dense.

Explain why the topological entropy of the cat map equals $\log\lambda_+$ via both the Markov partition and the Lyapunov-exponent route, and identify the general theorem they both invoke.

The Markov-partition route: the entropy of the subshift $\Sigma_A$ is $h_{\mathrm{top}}(\sigma|_{\Sigma_A})=\log\rho(A)$ (the standard subshift-of-finite-type entropy formula, by counting periodic points / the Perron–Frobenius theorem). The Markov partition is finite-to-one, so the entropy is preserved: $h_{\mathrm{top}}(f)=h_{\mathrm{top}}(\sigma)=\log\rho(A)$. The transition matrix of the cat map's two-rectangle partition has $\rho(A)=\lambda_+$ (the largest eigenvalue — the two matrices are related by a permutation and have the same spectral radius, which equals the expanding eigenvalue of the linear map $A$). The Lyapunov-exponent route: for a $C^{1+\alpha}$ diffeomorphism with an invariant measure $\mu$, the Ruelle inequality $h_\mu\le\int\sum_{\chi_i>0}\chi_i\,d\mu$; for the cat map with the Haar (Lebesgue) measure, the Lyapunov exponents are $\log|\lambda_\pm|$, the positive one is $\log\lambda_+$, and the Pesin formula gives equality $h_{\mathrm{Haar}}=\log\lambda_+$ (the Haar measure is the SRB measure, see Ergodic & Physical Measures). The variational principle $h_{\mathrm{top}}=\sup_\mu h_\mu$ then gives $h_{\mathrm{top}}=\log\lambda_+$. Both routes invoke the variational principle + the Pesin formula / the Markov coding, which agree for Anosov systems.

Pitfalls¶

Structural stability is $C^1$ (not $C^r$ for $r>1$ in general): the Anosov / Axiom A structural-stability theorem holds in the $C^1$ topology; the $C^r$ structural stability is more subtle and is not generic for $r>1$ (the Mañé $C^1$-stability conjecture, proven by Mañé / Hayashi, characterizes $C^1$-structurally-stable systems as Axiom A + no-cycles + transversality). The $C^r$ analog is open in general.
The shadowing / closing lemmas require uniform hyperbolicity: they fail for partial / non-uniform hyperbolicity; the non-uniform analog is the Pesin closing lemma, which requires $C^{1+\alpha}$ smoothness and controls the closing only on a set of full measure (not uniformly on $\Lambda$).
Markov partitions are for Axiom A, not for general hyperbolic sets: the Anosov–Bowen construction requires the spectral decomposition (basic sets); for a general hyperbolic set without the Axiom A density of periodic points, the Markov partition may not exist.
The semi-conjugacy $\pi:\Sigma_A\to\Lambda$ is finite-to-one, not one-to-one: the symbolic coding is not a conjugacy in general (the map $\pi$ identifies some orbits); the Bowen measure (max entropy) is the pushforward of the Parry measure on $\Sigma_A$, but the identification may collapse finitely many symbolic orbits.
Anosov flows have a 1-dim flow direction $E^0$ in the splitting: $T_\Lambda M=E^{ss}\oplus E^0\oplus E^{uu}$ (not $E^s\oplus E^u$ as for diffeomorphisms); the flow direction is neutral and is part of the structure (the flow is not an Anosov diffeomorphism).

Refs¶

Smale, Differentiable dynamical systems (BAMS 1967); Anosov, Geodesic flows on closed Riemann manifolds with negative curvature (Proc. Steklov 1967)
Katok–Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Hadamard–Perron, shadowing, Markov partitions, Axiom A)
Bowen, Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms (Markov partitions, spectral decomposition, entropy)
Mañé, A proof of the $C^1$ stability conjecture; Hayashi, Connecting invariant manifolds and the $C^1$-stability conjecture

Next¶

Hamiltonian & KAM · Bifurcations & Chaos · Ergodic & Physical Measures · DS index · Foundations · Ergodic theory · ET / Oseledets & Pesin — the Pesin formula / the Lyapunov / the SRB the ergodic realization of the hyperbolic · Topology · Symplectic geometry