Abstract
For each integer \(d\ge 2\), we identify the intersections of the connectedness locus of \(\overline{z}^{d}+c\) with the rays \(\omega \mathbb {R}^{+}\), where \(\omega ^{d+1}=\pm 1\).
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References
Baribeau L and Ransford T, Cross-sections of multibrot sets, J. Anal. 24(1) (2016) 95–101
Hubbard J H and Schleicher D, Multicorns are not path connected, Frontiers in complex dynamics, pp. 73–102, Princeton Math. Ser., 51 (2014) (Princeton: Princeton University Press)
Inou H, Self-similarity for the tricorn, Exp. Math. (2018) https://doi.org/10.1080/10586458.2017.1420502
Inou H and Mukherjee S, Non-landing parameter rays of the multicorns, Invent. Math. 204(3) (2016) 869–893
Lau E and Schleicher D, Symmetries of fractals revisited, Math. Intell., 18(1) (1996) 45–51
Milnor J, Remarks on iterated cubic maps, Exp. Math. 1(1) (1992) 5–24
Mukherjee S, Nakane S and Schleicher D, On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials, Ergod. Theory Dyn. Syst. 37(3) (2017) 859–899
Nakane S, Connectedness of the tricorn, Ergod. Theory Dyn. Syst. 13 (1993) 349–356
Nakane S and Schleicher D, On multicorns and unicorns I: antiholomorphic dynamics, hyperbolic components and real cubic polynomials, Int. J. Bifurc. Chaos 13 (2003) 2825–2844
Parisé P, Ransford T and Rochon D, Tricomplex dynamical systems generated by polynomials of even degree, Proceedings of Chaos 2016, Chaotic Modeling and Simulation (2017) no. 1, pp. 37–48
Parisé P and Rochon D, Tricomplex dynamical systems generated by polynomials of odd degree, Fractals, 25(3) (2017) 1750026 (11 pages)
Sheng X and Spurr M J, Symmetries of fractals, Math. Intell. 18(1) (1996) 35–42
Acknowledgements
The author would like to thank Fei Yang for his comments, remarks and careful reading of the text. Part of this work was supported by the National Natural Science Foundation of China (Grant No. 11801305). He would also like to thank the referee for helpful comments.
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Communicating Editor: Kaushal Verma
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Zhang, X. Cross-sections of the multicorns. Proc Math Sci 129, 28 (2019). https://doi.org/10.1007/s12044-019-0469-9
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DOI: https://doi.org/10.1007/s12044-019-0469-9