Expanding metrics for unicritical semihyperbolic polynomials

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 $ρ(z) |dz|$, where $ρ(z) = 1+\frac{1}{\textrm{dist}(z,P(f))^{1-1/d}}$, and where $P(f)$ is the postcritical set of $f$. We also show that this metric is Hölder equivalent to the usual Euclidean metric.

No reviews yet.

No comments yet.