@WillieWong Besides O'Neill, there is also the Book Einstein Manifolds of Arthur L. Besse.

I'd look at continued fraction algorithms

I read the two relevant statements (Lemmas 9 and 13) in the Nietecko-Shub paper and the proof of the second one. The proof has a few problems. Their constant 2pi is in fact the better constant 4 (actually this is good). They use state Lemma 9 with only one path but use it with several paths. The claim that the time-1 maps sends the slice 0 to the slice 2pi is likely wrong (this can be fixed but changes the estimate at the end ; maybe Lemma 9 can be adapted in fact).

It implies the result I want. But there is the distance complication. I will look in the article to see if they do the simple case but I doubt.

It was interesting reading this piece of history. Thanks. Too far from what I need, though.

Very nice result and reference. Almost what I am looking for.

Quite interesting, thanks. I will look closely.

Thanks. That approach seems overkill. Btw what is Guggenheim's theorem?

Good point @KevinCasto, I have edited the question

I think there can be resonances even in the case all eigenvalues are less than one in modulus. They can prevent linearization.