Fixed points of Koch's maps

TL;DR

This paper investigates the eigenvalues of fixed points of Koch's maps, confirming a conjecture that they are either zero or have modulus greater than one, and relates these values to polynomial multipliers.

Contribution

It proves that eigenvalues at fixed points are either zero or greater than one in modulus, confirming a conjecture and linking these to polynomial multipliers.

Findings

Eigenvalues at fixed points are either 0 or have modulus > 1.

Eigenvalues along periodic cycles outside post-critical sets have modulus > 1.

Eigenvalues at fixed points are explicitly described in terms of polynomial multipliers.

Abstract

We study endomorphisms constructed by Sarah Koch in her thesis and we focus on the eigenvalues of the differential of such maps at its fixed points. In Koch's thesis, to each post-critically finite unicritical polynomial, Koch associated a post-critically algebraic endomorphism of $\mathbb{CP}^k$. Koch showed that the eigenvalues of the differentials of such maps along periodic cycles outside the post-critical sets have modulus strictly greater than $1$. In this article, we show that the eigenvalues of the differentials at fixed points are either $0$ or have modulus strictly greater than $1$. This confirms a conjecture proposed by the author in his thesis. We also provide a concrete description of such values in terms of the multiplier of a unicritical polynomial.