On the connectivity of the Julia set of a finitely generated rational semigroup (Q2750885)

The set of nonconstant rational mappings on the Riemann sphere \(\overline{\mathbb{C}}\) is a semigroup under composition. Let \(G\) be its subsemigroup. The domain of normality for \(G\) is called the Fatou set \(F(G)\), and its complement \(\overline{\mathbb{C}} \setminus F(G)\) is called the Julia set \(J(G)\). The authors prove that the Julia set \(J(G)\) of a finitely generated rational semigroup \(G\) is connected if the union of the Julia sets of generators is contained in a subcontinuum of \(J(G)\). Under a nonseparating condition, the authors prove that the Julia set of a finitely generated polynomial semigroup is connected if its postcritical set is bounded.