Growth of attraction rates for iterates of a superattracting germ in dimension two

TL;DR

This paper investigates the growth patterns of attraction rates for iterates of superattracting fixed points in two complex dimensions, revealing linear recursion relations and stability conditions using valuative techniques.

Contribution

It introduces a method to analyze attraction rate sequences via valuative techniques, showing they satisfy linear recursions and establishing stability conditions for finite germs.

Findings

Attraction rate sequences satisfy an integral linear recursion relation.

The recursion relation can be simplified to order at most two after iteration.

Finite germs admit a bimeromorphic model with weak local algebraic stability.

Abstract

We study the sequence of attraction rates of iterates of a dominant superattracting holomorphic fixed point germ f:(C^2,0)->(C^2,0). By using valuative techniques similar to those developed by Favre-Jonsson, we show that this sequence eventually satisfies an integral linear recursion relation, which, up to replacing f by an iterate, can be taken to have order at most two. In addition, when the germ f is finite, we show the existence of a bimeromorphic model of (C^2,0) where f satisfies a weak local algebraic stability condition.