Indecomposable continua and the Julia sets of polynomial-like mappings

Abstract

Let \(f\) be a polynomial-like mapping of the sphere of degree \(d \geq 2\). We show that the Julia set \(J(f)\) of \(f\) cannot be the union of a finite number of proper indecomposable subcontinua. As a corollary, we prove that \(J(f)\) is an indecomposable continuum if and only if there exists a prime end of some complementary region of \(J(f)\) whose impression is the entire \(J(f)\), generalizing a result by Childers, Mayer and Rogers.