Mandelpinski spokes in the parameter planes of rational maps

摘要

In this paper we describe a new structure that arises in the parameter plane of the family of maps z(n) + lambda/z(d) where n >= 2 is even but d >= 3 is odd. We call these structures Mandelbrot-Sierpinski spokes (or, for short, 'Mandelpinski spokes'). It is known that there are infinitely many baby Mandelbrot sets in these parameter planes that are part of what is called the Mandelpinski maze for these maps. We show here that there are infinitely many 'spokes' emanating from each of these Mandelbrot sets. Each spoke consists of infinitely many alternating Mandelbrot sets and Sierpinski holes that lie along a certain arc that tends away from the given Mandelbrot set in a certain direction.