Fixed points of Koch's maps

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.