We consider all the transcendental meromorphic functions from the class S whose Julia set is a Jordan curve. We show that then the Julia set is either a straight line or its Hausdorff dimension is strictly larger than 1.

We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpiński curve.

We prove the following theorem about Julia sets of the maps fn,p,γ(z) = z n + γ zp , for integers n, p ≥ 2, γ ∈ C by using techniques developed for the Weierstrass elliptic ℘ function and adapted to this setting. Folklore connectedness theorem: If fn,p,γ has a bounded critical orbit, then J(fn,p,γ) is connected. This is related to connectivity results by the author and others about J(℘), where ...

In this note we give answers to questions posed to us by J. Milnor and M. Shub, which shed further light on the structure of non-computable Julia sets.

Responding to Change on TV: How Viewer-Controlled Changes in Content Differ From Programmed Changes in Content Kevin Wise ab; Seungjo Lee cd; Annie Lang ef; Julia R. Fox gh; Maria Elizabeth Grabe ij a Stanford University, b Missouri School of Journalism, c Indiana University, d Department of Mass Communications, Chungang University, e University of Wisconsin-Madison, f Telecommunications and Co...

We show that if P is a quadratic polynomial with a fixed Cremer point and Julia set J , then for any monotone map φ : J → A from J onto a locally connected continuum A, A is a

We give an example of two rational functions with non-equal Julia sets that generate a rational semigroup whose completely invariant Julia set is a closed line segment. We also give an example of polynomials with unequal Julia sets that generate a non nearly Abelian polynomial semigroup with the property that the Julia set of one generator is equal to the Julia set of the semigroup. These examp...

We prove a special case of the Dynamical André-Oort Conjecture formulated by Baker and DeMarco [3]. For any integer d ≥ 2, we show that for a rational plane curve C parametrized by (t, h(t)) for some non-constant polynomial h ∈ C[z], if there exist infinitely many points (a, b) ∈ C(C) such that both zd+a and zd+b are postcritically finite maps, then h(z) = ξz for a (d − 1)-st root of unity ξ. A...

We study stability questions for dynamics on Pn. To do so we use locally simply parameterized holomorphic families. We prove a weak version of the λ-lemma for dynamical systems on Pn. This takes the form of showing that the set of potential traces for a holomorphic motion that preserve the dynamics (i.e. the set of “Fatou sections”) is compact. We introduce the notion of a postcritically bounde...