Geometric localization, uniformly John property and separated semihyperbolic dynamics (Q1916302)

The authors investigate the existence of localizations for John domains and apply their results to the set \(A_\infty(f)\), the domain of attraction to \(\infty\) of a polynomial \(f\). Here are some of the results obtained: A John disk admits a localization, but an example is given of a John domain which does not admit a localization. For a semihyperbolic polynomial \(f\), the domain \(A_\infty(f)\) admits a localization if and only if \(f\) is separated semihyperbolic. Further, a John domain is localizable if and only if it is uniformly John. An example is given of a semihyperbolic polynomial which is not separated semihyperbolic. A final result shows that a localizable John domain has a ``doubling property'' relative to harmonic measure.