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Abstract: We study the topology of the Julia set of a quadratic Cremer polynomial P. Our main tool is the following topological result. Let be a homeomorphism of a plane domain U and let be a non-degenerate invariant non-separating continuum. If T contains a topologically repelling fixed point x with an invariant external ray landing at x, then T contains a non-repelling fixed point. Given P, two angles are K-equivalent if for some angles the impressions of and are non-disjoint, ; a class of K-equivalence is called a K-class. We prove that the following facts are equivalent: (1) there is an impression not containing the Cremer point; (2) there is a degenerate impression; (3) there is a full Lebesgue measure dense -set of angles each of which is a K-class and has a degenerate impression; (4) there exists a point at which the Julia set is connected im kleinen; (5) not all angles are K-equivalent. [Copyright &y& Elsevier]

Abstract: We find necessary and sufficient conditions for the connected Julia set of a polynomial of degree to be an indecomposable continuum. One necessary and sufficient condition is that the impression of some prime end (external ray) of the unbounded complementary domain of the Julia set J has nonempty interior in J. Another is that every prime end has as its impression the entire Julia set. The latter answers a question posed in 1993 by the second two authors. We show by example that, contrary to the case for a polynomial Julia set, the image of an indecomposable subcontinuum of the Julia set of a rational function need not be indecomposable. [Copyright &y& Elsevier]