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…)