Attraction rates for iterates of a superattracting skew product

TL;DR

This paper analyzes attraction rates in the dynamics of superattracting skew products, providing inequalities that apply broadly beyond just superattracting cases, extending previous work on invariant wedges and dominant terms.

Contribution

It introduces inequalities for attraction rates in skew product dynamics, generalizing previous results to all cases beyond superattracting fixed points.

Findings

Derived inequalities for vertical attraction rates

Extended applicability to non-superattracting cases

Built on previous construction of Böttcher coordinates

Abstract

Let $f(z,w)=(p(z),q(z,w))$ be a holomorphic skew product with a superattracting fixed point at the origin. In the previous paper we have succeeded to specify a dominant term of $q$ by the order of $p$ and the Newton polygon of $q$ and to construct a B\"{o}ttcher coordinate on an invariant wedge. By using the same idea and terminologies, we give inequalities of attraction rates for the vertical dynamics of $f$ in this paper. The results hold not only for the superattracting case, but for all the other cases.