An explicit bound for uniform perfectness of the Julia sets of rational maps

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Abstract

A compact set C in the Riemann sphere is called uniformly perfect if there is a uniform upper bound on the moduli of annuli which separate C. Julia sets of rational maps of degree at least two are uniformly perfect. Proofs have been given independently by Mañé and da Rocha and by Hinkkanen, but no explicit bounds are given. In this note, we shall provide such an explicit bound and, as a result, give another proof of uniform perfectness of Julia sets of rational maps of degree at least two. As an application, we provide a lower estimate of the Hausdorff dimension of the Julia sets. We also give a concrete bound for the family of quadratic polynomials fc(z) = z2+c in terms of the parameter c.

Original languageEnglish
Pages (from-to)317-333
Number of pages17
JournalMathematische Zeitschrift
Volume238
Issue number2
DOIs
Publication statusPublished - 2001 Oct

ASJC Scopus subject areas

  • Mathematics(all)

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