Link to this article: http://journals.cambridge.org/abstract_S0143385704000380 How to cite this article: PAUL BLANCHARD, ROBERT L. DEVANEY, DANIEL M. LOOK, PRADIPTA SEAL and YAKOV SHAPIRO (2005). Sierpinski-curve Julia sets and singular perturbations of complex polynomials.Abstract. In this paper we consider the family of rational maps of the complex plane given by z 2 + λ z 2 where λ is a complex parameter. We regard this family as a singular perturbation of the simple function z 2 . We show that, in any neighborhood of the origin in the parameter plane, there are infinitely many open sets of parameters for which the Julia sets of the corresponding maps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However, we also show that parameters corresponding to different open sets have dynamics that are not conjugate.
We define a locally Sierpinski Julia set to be a Julia set of an elliptic function which is a Sierpinski curve in each fundamental domain for the lattice. In order to construct examples, we give sufficient conditions on a lattice for which the corresponding Weierstrass elliptic ℘ function is locally connected and quadratic-like, and we use these results to prove the existence of locally Sierpinski Julia sets for certain elliptic functions. We give examples satisfying these conditions. We show this results in naturally occurring Sierpinski curves in the plane, sphere and torus as well.
Abstract. In this paper we prove the existence of a new type of Sierpinski curve Julia set for certain families of rational maps of the complex plane. In these families, the complementary domains consist of open sets that are preimages of the basin at ∞ as well as preimages of other basins of attracting cycles.In recent years the families of rational maps of the complex plane given by z n + λ/z d have been shown to exhibit a rich array of both dynamical and topological phenomena. The principal focus of these studies has most often been the Julia sets for such maps. As is well known, the Julia set is the set on which all of the "interesting" dynamics occurs. For many λ-values, the Julia sets of these maps are also quite interesting from a topological point of view.For example, for each n ≥ 2, it is known that there are infinitely many λ-values for which the Julia set is a generalized Sierpinski gasket (see [5]), and none of these Julia sets are homeomorphic to each other. As another example, there are infinitely many λ-values in each of these families for which the Julia set is a Sierpinski curve ([1]). A Sierpinski curve is any planar set that is compact, connected, locally connected, nowhere dense, and has the property that any two complementary domains are bounded by simple closed curves that are pairwise disjoint. A result of Whyburn [13] shows that any such set is homeomorphic to the well-known Sierpinski carpet fractal. The interesting topology arises from the fact that a Sierpinski curve is universal in the sense that it contains a homeomorphic copy of any planar, onedimensional continuum. In the case of the Sierpinski curve Julia sets, all of these sets are homeomorphic, but as shown in [4], infinitely many of them are dynamically distinct in the sense that the corresponding maps are not topologically conjugate on their Julia sets. Moreover, as shown in [1], when n = 2, d = 2 or n = 2, d = 1, in every neighborhood of the parameter value λ = 0, there are infinitely many parameter values for which the Julia set is a Sierpinski curve on which the dynamics are distinct. Hence these families undergo a dramatic explosion when λ becomes nonzero.In each of the above cases where the Julia set is a Sierpinski curve, the complementary domains (or the Fatou components) are always preimages of the immediate basin of attraction of ∞, which is a superattracting fixed point for these maps (provided n ≥ 2). In this paper, we exhibit a similar infinite collection of dynamically distinct Julia sets, but now the Fatou components are quite different. Instead of being preimages of a single superattracting basin at ∞, we give examples where the 2000 Mathematics Subject Classification. Primary: 37F45; Secondary: 37F20.
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