Fixed points of Koch maps

Abstract

We study endomorphisms constructed by Sarah Koch in [Teichmüller theory and critically finite endomorphisms, Advances in Mathematics 248 (2013), 573–617] and we focus on the eigenvalues of the differential of such maps at its fixed points. To each post-critically finite unicritical polynomial, Koch associated a post-critically algebraic endomorphism of C P k {\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 1 . In this article, we show that the eigenvalues of the differentials at fixed points are either 0 0 or have modulus strictly greater than 1 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.