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Immediate and Virtual Basins of Newton’s Method for Entire Functions

Sebastian Mayer, Dierk Schleicher (2006)

Annales de l’institut Fourier

We investigate the well known Newton method to find roots of entire holomorphic functions. Our main result is that the immediate basin of attraction for every root is simply connected and unbounded. We also introduce “virtual immediate basins” in which the dynamics converges to infinity; we prove that these are simply connected as well.

Intertwined internal rays in Julia sets of rational maps

Robert L. Devaney (2009)

Fundamenta Mathematicae

We show how the well-known concept of external rays in polynomial dynamics may be extended throughout the Julia set of certain rational maps. These new types of rays, which we call internal rays, meet the Julia set in a Cantor set of points, and each of these rays crosses infinitely many other internal rays at many points. We then use this construction to show that there are infinitely many disjoint copies of the Mandelbrot set in the parameter planes for these maps.

Introduction

Pascale Roesch (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

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