Buried Sierpinski curve Julia sets

Abstract

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.

Cantor sets of circles of Sierpiński curve Julia sets

“…In fact, there are infinitely many other copies of the Mandelbrot set in N (see [3]). In [6], the following was shown. This leads to another way that Sierpiński curves can occur as Julia sets in these families.…”

Real and complex behavior for networks of coupled logistic maps

“…For polynomial maps, it has been shown that the connectivity of a map's Julia set is tightly related to the structure of its critical orbits (i.e., the orbits of the map's critical points). Due to extensive work spanning almost one century, from Julia [8] and Fatou [9] until recent developments [10,11], we now have the following: For a single iterated logistic map [12,13], the Fatou-Julia Theorem implies that the Julia set is either totally connected, for values of c in the Mandelbrot set (i.e., if the orbit of the critical point 0 is bounded), or totally disconnected, for values of c outside of the Mandelbrot set (i.e., if the orbit of the critical point 0 is unbounded). In previous work, the authors showed that this dichotomy breaks in the case of random iterations of two maps [19].…”

A Cantor-Mandelbrot-Sierpiński tree in the parameter plane for rational maps

“…We begin by reviewing some basic properties of functions in this family. See [1] or [4] for proofs of these facts.…”