Expanding metrics for unicritical semihyperbolic polynomials

TL;DR

This paper proves that unicritical semihyperbolic polynomials are uniformly expanding on their Julia set using a specific metric, which is also H"older equivalent to the Euclidean metric, enhancing understanding of their dynamical properties.

Contribution

It introduces a new metric for unicritical semihyperbolic polynomials that demonstrates uniform expansion and establishes its H"older equivalence to the Euclidean metric.

Findings

Proves uniform expansion of these polynomials with respect to the new metric.

Shows the new metric is H"older equivalent to the Euclidean metric.

Provides a metric-based framework for analyzing polynomial dynamics.

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