Leaves of stable foliation of holomorphic Anosov diffeomorphism

Question

I'm trying to understand the first half of the paper "Holomorphic Anosov systems" by E. Ghys (the journal reference is Inventiones mathematicae volume 119, pages 585–614(1995)). My question is about a particular claim that Ghys makes. I am a finishing undergraduate, so I suspect my main issue is missing background.

The setup is a compact complex manifold $M$ equipped with a holomorphic Anosov diffeomorphism $\phi$, and it is assumed that the unstable foliation has complex dimension 1. In the Proof of Proposition 2.2 (in the penultimate sentence of the second paragraph), Ghys says that the leaves of the stable foliation are simply connected, and even diffeomorphic to some euclidean space. He later also uses this fact in the proof of his Theorem B.

I don't understand why this is true and I wasn't able to find some hint or explanation in Ghys' references. Is there some simple explanation, or some good reference, for this fact? Is it special to all of the assumptions I stated in the setup, or is it something more general?

Answer 1

The fact that a stable manifold is diffeomorphic to a (real) Euclidean space is a consequence of the Stable Manifold Theorem, see for instance [Katok&Hasselblatt, Introduction to the modern theory of dynamical systems. chap 6 sec 4]:

By the Stable Manifold Theorem for every $p\in M$, there is a local stable manifold $W^s_{loc}(p)$ which is diffeomorphic to an Euclidean ball, and the global stable manifold satisfies $W^s(p)=\bigcup_{n\geq 0} f^{-n}( W^s_{loc}(f^n(p)))$. Thus $W^s(p)$ is a monotone union of Euclidean balls, hence $W^s(p)$ itself is diffeomorphic to an Euclidean ball. Here is nothing to do with holomorphic map.

To determine the complex structure of a stable manifold when we have a holomorphic diffeomorphism, is more difficult. When $W^s(p)$ has dimension 1, it can be shown that $W^s(p)$ is biholomorphic to $\mathbb{C}$ but not $\mathbb{D}$, see for instance [Bedford&Smillie, Polynomial diffeomorphisms of $\mathbb{C}^2$: currents, equilibrium measure and hyperbolicity. Thm 5.4]. It is conjectured that if $f:M\to M$ is a holomorphic diffeomorphism with an invariant hyperbolic set $\Lambda$, then a $k-$dimensional stable manifold of $p\in\Lambda$ is biholomorphic to $\mathbb{C}^k$. As far as I known this is still open.