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The Hausdorff Dimension is Convex on the Left Side of 1/4

Part of: Complex dynamical systems

Published online by Cambridge University Press:  10 January 2017

Ludwik Jaksztas
Ludwik Jaksztas*
Affiliation:
Faculty of Mathematics and Information Sciences, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland ([email protected])
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Abstract

Let d(c) denote the Hausdorff dimension of the Julia set Jc of the polynomial fc (z) = z 2 +c. The function c ↦ d(c) is real-analytic on the interval (–3/4, 1/4), which is in the domain bounded by the main cardioid of the Mandelbrot set. We prove that the function d is convex close to 1/4 on the left side of it.

Keywords

Hausdorff dimensionJulia setquadratic family

MSC classification

Secondary: 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations 37F35: Conformal densities and Hausdorff dimension
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 4 , November 2017 , pp. 911 - 936
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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