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 1 Let
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
:
witha 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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