2020/01/30 by Tao Chen, Chen, Tao, Yunping Jiang +3
Mathematics · #30D30 #32A20 #37F10 #37F20 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Primary: 37F30 #Secondary: 30F30
paper · pdf · doi:10.48550/arxiv.2001.11454
openalex publication_date 2020/01/30 · openalex created_date 2020/02/07 · openalex updated_date 2026/07/28
In this paper we continue the study, began in \citeCJK2, 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 \citeFK, 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.