Uniform non-autonomous basins of attraction

TL;DR

This paper proves that the basin of a uniformly attracting family of holomorphic maps is biholomorphic to complex Euclidean space, confirming a conjecture about the structure of stable manifolds in complex dynamics.

Contribution

It establishes that all such basins are biholomorphic to complex Euclidean space, resolving a longstanding conjecture in complex dynamics.

Findings

Basin of a uniformly attracting family is biholomorphic to complex Euclidean space

Confirms the conjecture on the biholomorphism type of stable manifolds

Provides a new understanding of the structure of stable manifolds in holomorphic dynamics

Abstract

It has been conjectured that every stable manifold arising from a holomorphic automorphism, that acts hyperbolically on a compact invariant set, is biholomorphic to complex Euclidean space. Such stable manifolds are known to be biholomorphic to the basin of a uniformly attracting family of holomorphic maps. It is shown that the basin of a uniformly attracting family of holomorphic maps is biholomorphic to complex Euclidean space and this resolves the conjecture on the biholomorphism type of such stable manifolds affirmatively.