We now state and prove the ugly technical theorem invoked yesterday, that you can refine certain “approximate” solutions of conjugacy-like equations involving Anosov diffeomorphisms (and maps close to them—though actually one can prove that Anosov diffeomorphisms are open in the topology). The proof is rather complicated, but it will basically rely on familiar techniques: hyperbolic linearization (in Banach spaces!), the contraction principle, and simple algebraic manipulation.

Theorem 1Let be an Anosov diffeomorphism of the compact manifold . Then if is sufficiently small, there is satisfying the following condition. Suppose , and one has an “approximately commutative diagram” for a map :

with a topological space and a homeomorphism: i.e. . Then there is a unique continuous close to (namely ) such that the modified diagram

commutes exactly.

So, how are we going to prove this? First, we want some sort of linearity, but we can’t add two elements of a manifold. Thus, we use the Whitney embedding theorem to assume without loss of generality that is a closed submanifold of . We can use the tubular neighborhood of , say the neighborhood of radius . In other words, every can be written uniquely as where and is of length and orthogonal to .

With this embedding, it is possible to add vectors in , although obviously we’re not necessarily going to get another element of by doing so. Nevertheless, there is a retraction of : namely, we map . In particular, this means that we can extend the function to .

Now, we are interested in finding such that , i.e. ; we’ve written the problem as finding a fixed point of the map . But we also want to be close to . In particular, there is (we hope) a small map with small norm such that (in !) maps into and satisfies our conditions. So our equation becomes

Now, define the map

This map is defined for sufficiently small. More precisely, if we consider the Banach space of such maps with the sup norm, then this map is defined on a neighborhood of the origin in —this is because is defined in a neighborhood of .

If we find a fixed point for , then this vector field can be used to construct which will satisfy the equation , thus creating the map in question. Well—sort of. That is, we’d first need to see that actually takes values in ! But this is straightforward. Since by definition takes values in , the equation shows that so does . (Neat trick.)

Granted, it would be sweet if we could simply invoke the contraction principle onto . But we can’t. The problem is that the enlargement by may not be a contraction. So we have to work harder.

And this is where differential calculus in Banach spaces comes in handy—we will approximate by its linearization, which *does* have a fixed point. Indeed, let’s compute the derivative ; this is a straightforward exercise. If is a tangent vector, a tangent curve at in the space is given by . Now

which is

I claim now (taking ) that this linear map , defined for , satisfies the condition that is invertible. Once this is done, the theorem will be straightforward, as we will be able to construct a fixed point.

** Analysis of the linear operator **

We are going to show the invertibility of by another trick. We will decompose the Banach space into three pieces.

So, anyway, we’re going to consider each tangent vector (tangent to ) as a vector field mapping to a point of . Now there is a splitting of continuous bundles on ; we can extend this to a neighborhood and adding to get , at least if is really tiny.

This yields a (topological) splitting of Banach spaces corresponding to vector fields that take their values in the corresponding bundle. Anyway, we know (since is -close to ) that a high power of , say , induces a linear map of norm less than on , so by continuity this is also true for . Same for (with the inverse). On , is zero!

It follows that we have an expression in the following form:

Here are small since is -close to . Similarly, it follows that high powers of and have small norm. From this, It follows that is invertible; indeed, it can be checked that by splitting into its components. Even more strongly, if is taken close enough to , then is bounded *independent of *

.

** The fixed point **

We now still need to find a fixed point. Write . Here . To say that is to say that

i.e. . But . So if is restricted to a really small neighborhood of the origin, and is restricted to being correspondingly small (which it will be if is -close enough to ), then this map

will be a map of balls of size in . It is indeed a contraction since is and has a bounded inverse. Which means we can apply the contraction principle to this guy and deduce a fixed point, hence a fixed point for —and hence, the big theorem!

I have followed a proof of Anatole Katok (who, incidentally, is at PSU) presented in the book of Brin and Stuck on dynamical systems.

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