Dynamics of McMullen maps (Q765660)

For \(n \geq 3\) and \(\lambda \in \mathbb{C}\setminus\{0\}\) let \(f_\lambda(z)=z^n + \lambda/z^n\), and denote by \(B_\lambda\) the immediate basin of infinity. Using Yoccoz puzzle techniques, the authors answer a question of Devaney by showing: Theorem 1.1 If the Julia set \(J(f_\lambda)\) is not a Cantor set, then the boundary \(\partial B_\lambda\) is a Jordan curve. Another result establishes regularity of \(\partial B_\lambda\) under natural dynamical hypotheses: Theorem 1.2 If \(J(f_\lambda)\) is not a Cantor set, then \(\partial B_\lambda\) is a quasi-circle if it contains neither parabolic nor recurrent critical points. Further results give conditions under which \(J(f_\lambda)\) is locally connected. The proofs makes use of a local version of the notion of ``semi-hyperbolicity'', called here the BD-condition.