Abstract
In this paper, we continue the study, began in Chen et al. (Slices of parameter space for meromorphic maps with two asymptotic values, arXiv:1908.06028, 2019), 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 Fagella and Keen (Stable components in the parameter plane of meromorphic functions of finite type, arXiv:1702.06563, 2017) and Chen and Keen (Discrete and Continuous Dynamical Systems 39(10):5659–5681, 2019), 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.
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Notes
Note that conjugating by the affine map \(w=z-q_0/2\), we obtain a map of the form \(\alpha \tan w\) for some \(\alpha \), \(|\alpha |>1\), with fixed points at \(\pm q_0/2\). In particular, if \(\rho _0\) is real, \(\lambda _0\) can be chosen real, and then the attracting basin of \(q_0\) is a half plane and the Julia set is a line.
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This material is based on work supported by the National Science Foundation. It is partially supported by a collaboration grant from the Simons Foundation (grant number 523341) and PSC-CUNY awards.
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Chen, T., Jiang, Y. & Keen, L. Accessible Boundary Points in the Shift Locus of a Family of Meromorphic Functions with Two Finite Asymptotic Values. Arnold Math J. 8, 147–167 (2022). https://doi.org/10.1007/s40598-020-00169-1
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DOI: https://doi.org/10.1007/s40598-020-00169-1