Expanding metrics for unicritical semihyperbolic polynomials

Abstract

We prove that unicritical polynomials $f(z)=z^d+c$ which are semihyperbolic, i.e., for which the critical point $0$ is a non-recurrent point in the Julia set, are uniformly expanding on the Julia set with respect to the metric $\rho (z)|dz|$, where $\rho (z)=1+\frac{1}{\text{dist}(z,P(f){)}^{1-1/d}}$, and where $P(f)$ is the postcritical set of $f$. We also show that this metric is Hölder equivalent to the usual Euclidean metric.

14 pages, 1 figure; simplified and streamlined the proof of Theorem 12, several other small changes and corrections of minor errors