On the dynamics of the one parameter functions \(F_ a (z)= z^ 2+ 2a \overline {z}\) (Q2563702)

The discrete dynamical system, determined in the complex plane by the function \(F_a(z)= z^2+2a \overline z\), \((a\in\mathbb{R}\) is a parameter) is investigated. Firstly, the singular set \(\Sigma_a\) of \(F_a(z)\) is the circle \(|z|= |a|\). Let \(\rho\) be a cube root of unity. The author proves that the point \(\infty\) is an attractive fixed point and if \(a \neq 0\), then the singular points \(z\in\Sigma_a \backslash \{a,\rho a,\rho^2a\}\) are folds and the points \(z\in \{a,\rho a, \rho^2a\}\) are cusps, hence \(F_a(\Sigma_a)\) is a hypocycloid of three cusps. Further, let \(K(F_a)= \{z\in\mathbb{C}; |F^n_a(z) |\) is bounded for all \(n\}\). Then the set \(K(F_a)\) is connected for \(a\in[-1,2]\) and is disconnected for \(a>2\) or \(a<-1\). The stable and unstable manifolds of fixed points and others points are investigated with respect to its relations to the Julia set \(J(F_a)\). Finally it is shown, that the family of maps \(F_a(z)= z^2+2a\overline z\), \(a\in\mathbb{C}\), is a universal unfolding of the map \(F_0(z)= z^2\).