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 \(\rho (z) = 1+{{\,\mathrm{dist}\,}}(z,P(f))^{-1+1/d}\) has singularities on the postcritical set P(f). We also show that this metric is Hölder equivalent to the usual Euclidean metric.
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Notes
For \(z_1 \in G\) there is a unique external ray, but we allow \(z_1 \in \partial G\), in which case there might be several.
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The author would like to thank the anonymous referee for valuable comments and suggestions.
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Geyer, L. Expanding Metrics for Unicritical Semihyperbolic Polynomials. Bull Braz Math Soc, New Series 52, 721–737 (2021). https://doi.org/10.1007/s00574-020-00228-3
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DOI: https://doi.org/10.1007/s00574-020-00228-3