Superstable manifolds of invariant circles and codimension-one Böttcher functions

Abstract

Abstract. Let f : X X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n > 1. Suppose there is an embedded copy of P 1 that is invariant under f , with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose f restricted to this line is given by z → z b , with resulting invariant circle S. We prove that if a ≥ b, then the local stable manifold W s loc (S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a ≥ b cannot be relaxed without adding additional hypotheses by presenting two examples with a < b for which W s loc (S) is not real analytic in the neighborhood of any point.

Lucjan Emil Böttcher and his Mathematical Legacy

“…It should be mentioned that the classical method of proving Böttcher's theorem was used recently in [21] to give the classification (up to holomorphic conjugacy) of attracting rigid germs in C2 (which was subsequently applied in [43]), and in [27] in the study of local stable manifolds of a dominant meromorphic self-map f : X X, where X is a compact Kähler manifold of dimension n > 1. As far as other applications and generalizations are concerned, see e.g.…”

Lucjan Emil Böttcher and his mathematical legacy

“…It should be mentioned that the classical method of proving Böttcher's theorem was used recently in [21] to give the classification (up to holomorphic conjugacy) of attracting rigid germs in C2 (which was subsequently applied in [43]), and in [27] in the study of local stable manifolds of a dominant meromorphic self-map f : X X, where X is a compact Kähler manifold of dimension n > 1. As far as other applications and generalizations are concerned, see e.g.…”