Abstract
We study the dynamics of a parameterized family of quadratic rational functions z 7→�z/(1−z) 2 , � ∈ (0,∞). We have investigated the dynamical and geometric properties of Julia sets and the nature of the variance of these properties according to the parameter. We have shown the Julia sets are connected when � > 1 and are dynamically defined cantor sets when � < 1. We also have shown the continuity of geometric properties specially in terms of Hausdorff dimensions of these Julia sets.
Highlights
IntroductionKoebe function is most celebrated function in geometric function theory
In this paper we consider the family Kλ(z) = (1 λz −z )2When λ=1, we get the famous Koebe function
We describe the complete dynamics of a larger family of rational maps containing Koebe function as a member and show the continuity of the Julia sets and their Hausdorff dimensions
Summary
Koebe function is most celebrated function in geometric function theory It is instrumental in proving many important results. We describe the complete dynamics of a larger family of rational maps containing Koebe function as a member and show the continuity of the Julia sets and their Hausdorff dimensions. Milnor [7, 8] and Rees [9] have investigated the moduli space of quadratic rational maps in general. The concept of conformal measures is one of the main tools to understand fractal properties of a Julia set, i.e. its dimensions and measures. Basic results concerning conformal measures, Hausdorff dimension and fractal and ergodic properties of Julia sets of rational functions of the Riemann sphere can be found in a survey done by Urbanski [13]
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