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.

中文翻译：

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单临界半双曲多项式的扩展度量

我们证明了单临界多项式 $f(z)=z^d+c$ 是半双曲线的，即临界点 $0$ 是 Julia 集中的非循环点，在 Julia 集上一致扩展到度量 $\rho(z) |dz|$，其中 $\rho(z) = 1+\frac{1}{\textrm{dist}(z,P(f))^{1-1/d }}$，其中 $P(f)$ 是 $f$ 的后临界集。我们还表明，这个度量比通常的欧几里得度量更旧。