I now know what I’m working on for my REU project; I’ll be studying (with two other undergraduates) a type of cohomology for dynamical systems. Misha Guysinsky, our mentor, has not explained the specific problem yet—perhaps that’ll come when we meet with him on Thursday. So I’ve spent the last weekend trying to learn a few basic facts about (especially hyperbolic) dynamical systems, which I will try to explain here.

1. Why do we care about hyperbolicity?

So, first a definition: let ${f: M \rightarrow M}$ be a ${C^1}$-morphism of a smooth manifold ${M}$. Suppose ${p \in M}$ is a fixed point. Then ${p}$ is called hyperbolic if the derivative ${Df_p: T_p(M) \rightarrow T_p(M)}$ has no eigenvalues on the unit circle. This comes from linear algebra: an endomorphism of a vector space is called hyperbolic if its eigenvalues are off the unit circle. Hyperbolicity is an important condition in dynamics, and I want to illustrate this with a few examples.

The first observation is that if we wiggle ${f}$ slightly, then the wiggled function still has a fixed point near ${p}$. This is a special case of structural stability for hyperbolic dynamical systems, and can be proved using a little bit of transversality.

So, first, I should explain what I mean by “wiggle.” I mean that if ${g: M \rightarrow M}$ is close to ${f}$ in the ${C^1}$-topology, then ${g}$ has a fixed point near ${p}$. To see this, draw the graph of ${f}$ as a submanifold of ${M \times M}$. Then the fact that ${Df}$ has no eigenvalues on the unit circle means that this graph is transversal to the diagonal submanifold ${\{(m,m)\} \subset M \times M}$ at ${p}$. Now suppose you have two submanifolds ${N_1, N_2 \subset N}$ which intersect transversally at a point ${p}$. Then if you wiggle ${N_1, N_2}$ slightly, then they will still intersect transversally near ${p}$; this is a basic fact from differential topology (cf. for instance the book of Guillemin and Pollack). Wiggling ${f}$ to ${g}$ corresponds to shifting the graph submanifold slightly, so the graph of ${g}$ must intersect the diagonal too near ${p}$. In particular, ${g}$ has a fixed point.

But there’s so much more.

2. Hyperbolic linear maps

The previous observation can be generalized to a useful fact about smooth manifolds in general. However, the notion of hyperbolic fixed point is inherently local—not every point is a fixed point! We need a global analog, that preserves the idea of hyperbolicity. To motivate this, let us begin with a little linear algebra.

The tangent map ${Df_p}$ can be thought of as the linear approximation to ${f}$ near ${p}$, so the first step is to talk a little more about hyperbolic linear maps. So let ${T: V \rightarrow V}$ by hyperbolic and invertible, for ${V}$ a (real) vector space.

Then there is a splitting

$\displaystyle \boxed{V = V_s \oplus V_u}$

where ${V_s}$ (${s}$ for stable—as will be explained below) consists of the points ${v \in V}$ such that ${T^i v \rightarrow 0}$ as ${i \rightarrow \infty}$, and ${V_u}$ of the points ${v}$ such that ${T^{-i}v \rightarrow 0}$ as ${i \rightarrow \infty}$. This can be seen by complexifying and taking ${V_{\mathbb{C}} = V \otimes_{\mathbb{R}} \mathbb{C}}$. On this (complex) space, we have a splitting into generalized eigenspaces corresponding to each eigenvalue of ${T}$.

Let ${V_{\mu}}$ be a generalized eigenspace where ${|\mu|<1}$. I claim that ${T^i \rightarrow 0}$ on ${V_{\mu}}$. Indeed, we can put ${T}$ in a Jordan normal form with ${\mu}$‘s on the diagonal and a few one’s on the second diagonal, and zeros everywhere else. One checks directly that ${T^i \rightarrow 0}$. Similarly, on the generalized eigenspaces with ${|\mu|>1}$, one checks that ${T^{-i} \rightarrow 0}$.

For ${v \in V - (V_s \cup V_u)}$, the stable and unstable components are both nonzero, and is easy to see that ${T^i v \rightarrow \infty}$ as ${i \rightarrow \pm \infty}$.

This means that the orbits of ${T}$ on ${V}$ are in the following forms: ${\{0\}}$, something starting near 0 and tending to ${\infty}$, something starting at ${\infty}$ and tending to zero, and something tending to ${\infty}$ either way.

This is a result for linear maps. For hyperbolic maps of manifolds, there is in fact such a result—called the stable manifold theorem, and much harder to prove.

3. Hyperbolic sets

Now suppose we have a morphism ${f: M \rightarrow M}$. We want to define this idea of hyperbolicity globally. Fix a subset ${\Lambda \subset M}$ on which ${f}$ is an immersion; we say that ${f}$ is hyperbolic on ${\Lambda}$ if ${f(\Lambda) \subset \Lambda}$ and the following condition is satisfied.

Instead of a splitting of vector spaces, we have a splitting of continuous vector bundles

$\displaystyle TM = TM^s \oplus TM^u$

which are invariant under ${Df}$ (i.e. ${Df(TM^s) \subset TM^s}$, etc.), and such that these satisfy the the following global analog of the hyperbolicity condition. That is, we assume there exists a Riemannian metric on ${TM}$ such that, with respect to the induced norm, we have for some ${0 < \lambda < 1}$,

$\displaystyle ||Df^n||_{TM^s} \leq C \lambda^n||v||$

$\displaystyle ||Df^{-n}||_{TM^u} \leq C \lambda^n||v||.$

The first condition, for instance, means that if ${v \in T_pM^s}$, then the norm of ${Df^n(v) \in T_{f^n(p)}}$ is at most ${C \lambda^n}$ times that of ${v}$. By analogy, ${TM^s, TM^u}$ are called the stable and unstable bundles, respectively.

This is a generalization of the usual condition of a hyperbolic fixed point, to which this reduces when ${\Lambda}$ has one point (by a little linear algebra). At the opposite extreme, when ${\Lambda}$ is the whole space ${M}$, we call ${f}$ an Anosov diffeomorphism. I’m mainly interested in talking about Anosov diffeomorphisms. However, it’s cool to keep a little extra generality when we can. This is why we want to consider hyperbolic sets.

4. The refinement lemma: How to construct topological conjugacies

In dynamical systems theory, we often want to clasisfy things up to topological conjugacy. This is a way of saying that structually, two dynamical systems are the same. In particular, a topological conjugacy between two maps ${f: X \rightarrow X, g: Y \rightarrow Y}$ (i.e., discrete dynamical systems) is given by a homeomorphism ${h: Y \rightarrow X}$ such that the following diagram commutes:

This means, in particular, that the orbits (and fixed points) of ${f,g}$ are in bijection, and much more—for instance, ${f,g}$ have the same topological entropy. They are basically the “same” dynamical system structurally. Diffeomorphism, incidentally, would be too strong for this to be nice. I asked a question some time back about this, and it got some interesting examples that give examples of conjugacies on the real line (and a useful trick); I don’t want to go into that here now though.

Anyway, this is why we define a discrete dynamical system to be structurally stable if any dynamical system “close” to it (in various topologies, often the ${C^1}$ one) is topologically conjugate. We will show that Anosov diffeomorphisms are structurally stable.

To do this, we will need a theorem that allows us to get exact solutions to commutative diagrams from “approximately commutative” diagrams; I think of it kind of as an analytic version of Hensel’s lemma. (Perhaps this says something about me. I’m pretty sure others think of Hensel’s lemma as an algebraic version of this sort of business…)

Here is the theorem we shall (someday) prove:

Theorem 1 Let ${f}$ be an Anosov diffeomorphism of the compact manifold ${M}$. There are ${\epsilon,\delta>0}$ satisfying the following condition. Suppose ${d_{C^1}(f,g)<\epsilon}$, and one has an “approximately commutative diagram” for ${\phi: X \rightarrow U}$

with ${X}$ a topological space and ${h: X \rightarrow X}$ a homeomorphism: i.e. ${d(g \circ \phi, \phi \circ h)< \epsilon}$. Then there is a unique continuous ${\psi: X \rightarrow U}$ close to ${\phi}$ (namely ${d(\psi, \phi)<\delta}$) such that the modified diagram

commutes exactly.

Note that the theorem is not actually about ${f}$, but about maps that are close to ${f}$!

There is a stronger theorem for hyperbolic sets, but I’m getting lazy here. You can look it up in Brin and Stuck’s book. Anyway, the statement of the genral theorem has far too many quantifers to be blog-suitable.

This is a fairly difficult theorem to prove. But the consequences are manifold. I’m going to defer the proof until later, and start discussing the consequences.

5. Structural stability for Anosov diffeomorphisms

Theorem 2 Anosov diffeomorphisms on commpact manifolds are structurally stable.

Well, ok. So first of all, fix our compact manifold ${M}$ and Anosov diffeomorphism ${f}$. Take ${\epsilon, \delta}$ as in the previous theorem. Now I claim that any diffeomorphism ${g: M \rightarrow M}$ such that ${d_{C^1}(f,g) < \epsilon}$ is topologically conjugate to ${f}$.

So, the diagram below is “approximately commutative;”

This means that we can find ${\psi: M \rightarrow M}$ close to the identity (and unique for that choice) such that ${f \circ \psi = \psi \circ g}$. We can do the same with ${f,g}$ reversed—the diagram is still approximately commutative, and our approximation lemma applies to approximations to Anosov diffeomorphisms—so that there is a unique ${\psi'}$ near the identity with ${\psi' \circ f = g \circ \psi'}$.

Now we have

$\displaystyle f \circ( \psi \circ \psi') = \psi \circ g \circ \psi' = (\psi \circ \psi') \circ f.$

The uniqueness portion of the approximation theorem applied to the (exactly!) commutative diagram with the identity map and ${f}$,

implies that ${\psi \circ \psi' = 1}$. Same for the other composition, and they are both homeomorphisms; they form a topological conjugacy between ${f,g}$.