On the family of cubic parabolic polynomials

Abstract

<p style='text-indent:20px;'>For a sequence <inline-formula><tex-math id="M1">\begin{document}$ (a_n) $\end{document}</tex-math></inline-formula> of complex numbers we consider the cubic parabolic polynomials <inline-formula><tex-math id="M2">\begin{document}$ f_n(z) = z^3+a_n z^2+z $\end{document}</tex-math></inline-formula> and the sequence <inline-formula><tex-math id="M3">\begin{document}$ (F_n) $\end{document}</tex-math></inline-formula> of iterates <inline-formula><tex-math id="M4">\begin{document}$ F_n = f_n\circ\dots\circ f_1 $\end{document}</tex-math></inline-formula>. The Fatou set <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{F}_0 $\end{document}</tex-math></inline-formula> is the set of all <inline-formula><tex-math id="M6">\begin{document}$ z\in\hat{\mathbb{C}} $\end{document}</tex-math></inline-formula> such that the sequence <inline-formula><tex-math id="M7">\begin{document}$ (F_n) $\end{document}</tex-math></inline-formula> is normal. The complement of the Fatou set is called the Julia set and denoted by <inline-formula><tex-math id="M8">\begin{document}$ \mathcal{J}_0 $\end{document}</tex-math></inline-formula>. The aim of this paper is to study some properties of <inline-formula><tex-math id="M9">\begin{document}$ \mathcal{J}_0 $\end{document}</tex-math></inline-formula>. As a particular case, when the sequence <inline-formula><tex-math id="M10">\begin{document}$ (a_n) $\end{document}</tex-math></inline-formula> is constant, <inline-formula><tex-math id="M11">\begin{document}$ a_n = a $\end{document}</tex-math></inline-formula>, then the iteration <inline-formula><tex-math id="M12">\begin{document}$ F_n $\end{document}</tex-math></inline-formula> becomes the classical iteration <inline-formula><tex-math id="M13">\begin{document}$ f^n $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M14">\begin{document}$ f(z) = z^3+a z^2+z $\end{document}</tex-math></inline-formula>. The connectedness locus, <inline-formula><tex-math id="M15">\begin{document}$ M $\end{document}</tex-math></inline-formula>, is the set of all <inline-formula><tex-math id="M16">\begin{document}$ a\in\mathbb{C} $\end{document}</tex-math></inline-formula> such that the Julia set is connected. In this paper we investigate some symmetric properties of <inline-formula><tex-math id="M17">\begin{document}$ M $\end{document}</tex-math></inline-formula> as well.</p>