TY - JOUR

T1 - An explicit bound for uniform perfectness of the Julia sets of rational maps

AU - Sugawa, Toshiyuki

N1 - Copyright:
Copyright 2005 Elsevier Science B.V., Amsterdam. All rights reserved.

PY - 2001/10

Y1 - 2001/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0035632994&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035632994&partnerID=8YFLogxK

U2 - 10.1007/s002090100255

DO - 10.1007/s002090100255

M3 - Article

AN - SCOPUS:0035632994

VL - 238

SP - 317

EP - 333

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 2

ER -