Adaptive trains for attracting sequences of holomorphic automorphisms

TL;DR

This paper improves the understanding of stable manifolds of hyperbolic holomorphic automorphisms by adapting conjugation techniques to the sequence of maps, especially under diagonal assumptions, leading to better biholomorphic equivalences.

Contribution

It introduces an adaptive method for conjugating sequences of automorphisms, enhancing previous results on stable manifold biholomorphisms, particularly when maps have diagonal linear parts.

Findings

Improved conjugation results for sequences of automorphisms.

Enhanced biholomorphic equivalence of stable manifolds to Euclidean space.

Applicability of the method in non-diagonal settings.

Abstract

Consider a holomorphic automorphism acting hyperbolically on an invariant compact set. It has been conjectured that the arising stable manifolds are all biholomorphic to Euclidean space. Such a stable manifold is always equivalent to the basin of a uniformly attracting sequence of maps. The equivalence of such basins to Euclideans has been shown under various additional assumptions. Recently Majer and Abbondandolo achieved new results by non-autonomously conjugating to normal forms on larger and larger time intervals. We show here that their results can be improved by adapting these time intervals to the sequence of maps. Under the additional assumption that all maps have linear diagonal part the adaptation is quite natural and quickly leads to significant improvements. We show how this construction can be emulated in the non-diagonal setting.