QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 5, 337–359 (2004) ARTICLE NO. 87 Cantor Necklaces and Structurally Unstable Sierpinski Curve Julia Sets for Rational Maps * Robert L. Devaney Boston University E-mail: bob@bu.edu In this paper we consider families of rational maps of degree 2n on the Riemann sphere F λ : C → C given by F λ (z)= z n + λ z n where λ ∈ C -{0} and n ≥ 2. One of our goals in this paper is to describe a type of structure that we call a Cantor necklace that occurs in both the dynamical and the parameter plance for F λ . Roughly speaking, such a set is homeomorphic to a set constructed as follows. Start with the Cantor middle thirds set embedded on the x-axis in the plane. Then replace each removed open interval with an open circular disk whose diameter is the same as the length of the removed interval. A Cantor necklace is a set that is homeomorphic to the resulting union of the Cantor set and the adjoined open disks. The second goal of this paper is to use the Cantor necklaces in the parameter plane to prove the existence of several new types of Sierpinski curve Julia sets that arise in these families of rational maps. Unlike most examples of this type of Julia set, the maps on these Julia sets are structurally unstable. That is, small changes in the parameter λ give rise to Julia sets on which the dynamical behavior is quite different. In addition, we also describe a new type of related Julia set which we call a hybrid Sierpinski curve. Key Words: Cantor necklace, Julia set, Sierpinski curve In this paper we consider families of rational maps of degree 2n on the Riemann sphere F λ : C → C given by F λ (z)= z n + λ z n where λ ∈ C \{0} and n ≥ 2. The case where n = 1 is very different and hence is excluded from this study. We denote the family of all such rational * Partially supported by NSF grant 0205779. 337