Dynamics of McMullen maps
نویسندگان
چکیده
In this article, we develop the Yoccoz puzzle technique to study a family of rational maps termed McMullen maps. We show that the boundary of the immediate basin of infinity is always a Jordan curve if it is connected. This gives a positive answer to the question of Devaney. Higher regularity of this boundary is obtained in almost all cases. We show that the boundary is a quasi-circle if it contains neither a parabolic point nor a recurrent critical point. For the whole Julia set, we show that the McMullen maps have locally connected Julia sets except in some special cases. Published by Elsevier Inc.
منابع مشابه
Frontiers in complex dynamics
Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complexanalytic, so a broad spectrum of techniques can contribute to their study (quasiconformal mappings, potential theory, algebraic geometry, etc.) The rational maps of a given degree form a finite dimensional manifold, so exploration of this parameter ...
متن کاملTHE HAUSDORFF DIMENSION OF THE BOUNDARY OF THE IMMEDIATE BASIN OF INFINITY OF McMULLEN MAPS
In this paper, we give a formula of the Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps fp(z) = z Q + p/z, where Q ≥ 3 and p is small. This gives a lower bound of the Hausdorff dimension of the Julia sets of McMullen maps in the special cases.
متن کاملTopology Proceedings 32 (2008) pp. 301-320: Evolution of the McMullen Domain for Singularly Perturbed Rational Maps
In this paper we study the dynamics of the two parameter family of complex maps given by fλ,a(z) = z n + λ/(z−a) where n ≥ 2 and d ≥ 1 are integers and a and λ are complex parameters. It is known that if a = 0 and |λ| 6= 0 is sufficiently small, the Julia set of fλ,a is a Cantor set of simple closed curves; this is the McMullen domain in the λ-plane. As soon as a 6= 0 (and |a| 6= 1), the Julia ...
متن کاملHausdorff dimension and conformal dynamics III: Computation of dimension
This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Dimension graphs are presented for (a) the family of Fuchsian groups generated by reflections in 3 ...
متن کامل