Stable manifolds of holomorphic diffeomorphisms

Abstract

We consider stable manifolds of a holomorphic diffeomorphism of a complex manifold. Using a conjugation of the dynamics to a (non-stationary) polynomial normal form, we show that typical stable manifolds are biholomorphic to complex Euclidean space.Comment: 17 pages. Revised version. To appear in Inv. Mat

Non-negatively curved Kähler manifolds with average quadratic curvature decay

“…where |H i (z)| ≤ C|z| for some constant C independent of i on D( By this lemma and Theorem 2.6, using the method in [9, §5], see also [19,30], we can prove the following: …”

Analytic invariant manifolds for sequences of diffeomorphisms

“…As in Section 2.1, we denote by B(E) ⊂ E the unit ball centered at zero, and by Δ(E) ⊂ E the polydisk in (6). Let now Y be the space of sequences (ϕ m ) m∈Z of analytic functions ϕ m = (ϕ 1m , ϕ 2m ) : B(E) → F 1 × F 2 , m ∈ Z, with a holomorphic extensionφ m to the interior of the polydisk Δ(E) which is continuous on Δ(E), and such that for every m ∈ Z, ϕ m (0) = 0, d 0 ϕ m = 0, and…”

“…The measure μ was introduced by Sibony (see [3,4]) and it was shown to be the unique measure of maximal entropy in [3]. The more general case of arbitrary holomorphic diffeomorphisms of a complex manifold was considered by Jonsson and Varolin in [6], where they showed that for each Lyapunov regular trajectory the stable and unstable manifolds are complex manifolds biholomorphic to a complex Euclidean space.…”

Méthodes de changement d’échelles en analyse complexe

“…La dynamique holomorphe apporte unéclairage nouveau sur les phénomènes de FatouBieberbach, le lecteur en trouvera un exposé dans le livre de Sibony [66]. La structure des variétés stables aétéétudiée par Jonsson et Varolin [49], nous n'avons présenté que le cas uniformément hyperbolique qui est particulièrement simple. Leur travail concerne surtout le cas non uniformément hyperbolique qu'ils abordent en combinant l'approche de Sternberg citée plus haut et la théorie de Pesin.…”