Abstract
We consider a non-conformal cubic map given by \(F_c(z)= z^3+c{\bar{z}}\) for c real. We study some properties of filled Julia set of this map denoted by \(K_{F_c}\) and show that if \(|c|<3/4\) then \(K_{F_c}\) is connected. We also show that the map \(F_c(z)\) has relation not only to the cubic family \(f_c(z)=z^3+cz\), but also to an important holomorphic map on complex projective space \({\mathbb {C}}{\mathbb {P}}^2\) given by \((z:w:\eta )\mapsto (z^3+cw\eta ^2: w^3+cz \eta ^2:\eta ^3)\) with \(\eta \ne 0\).
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Acknowledgements
The first author was partially supported by FAPEMIG APQ-02375-21, RED-00133-21. The forth author was partially supported by a Guilan university research Grant with process number 00764-11. He also wishes to thank the University of Guilan (Iran) for support, invitation and their hospitality.
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Alvez, A.M., Machado, D.S., Mehdipour, P. et al. On a Non-conformal Cubic Family and Its Application. Qual. Theory Dyn. Syst. 22, 152 (2023). https://doi.org/10.1007/s12346-023-00852-6
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DOI: https://doi.org/10.1007/s12346-023-00852-6