Accessible Boundary Points in the Shift Locus of a Familiy of Meromorphic Functions with Two Finite Asymptotic Values
Abstract
In this paper we continue the study, began in CJK2, of the bifurcation locus of a family of meromorphic functions with two asymptotic values, no critical values and an attracting fixed point. If we fix the multiplier of the fixed point, either of the two asymptotic values determines a one-dimensional parameter slice for this family. We proved that the bifurcation locus divides this parameter slice into three regions, two of them analogous to the Mandelbrot set and one, the shift locus, analogous to the complement of the Mandelbrot set. In FK, CK it was proved that the points in the bifurcation locus corresponding to functions with a parabolic cycle, or those for which some iterate of one of the asymptotic values lands on a pole are accessible boundary points of the hyperbolic components of the Mandelbrot-like sets. Here we prove these points, as well as the points where some iterate of the asymptotic value lands on a repelling periodic cycle, are also accessible from the shift locus.