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 $C^1$ 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 ${f}$ be an Anosov diffeomorphism of the compact manifold ${M}$. Then if ${\delta>0}$ is sufficiently small, there is ${\epsilon>0}$ satisfying the following condition. Suppose ${d_{C^1}(f,g)<\epsilon}$, and one has an “approximately commutative diagram” for a map $\phi: X \to M$:

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.

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 ${M}$ is a closed submanifold of ${\mathbb{R}^N}$. (more…)