Daniel M. Look

Sierpinski-curve Julia sets and singular perturbations of complex polynomials

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 correspondingmaps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However,we also showthatparameterscorrespondingto differentopensets havedynamics that are not conjugate.

LOCALLY SIERPINSKI JULIA SETS OF WEIERSTRASS ELLIPTIC ℘ FUNCTIONS

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.

Symbolic dynamics for a Sierpinski curve Julia set

In this paper we investigate the dynamics of certain rational functions on their Julia sets. We pay particular attention to the case where the Julia set is a Sierpinski curve. In this case, any two such Julia sets are known to be homeomorphic. However, the dynamics on these sets are often quite different. In this paper, we use symbolic dynamics to show how these different dynamical behaviors may be analyzed.

CHECKERBOARD JULIA SETS FOR RATIONAL MAPS

In this paper, we consider the family of rational maps where n ≥ 2, d ≥ 1, and λ ∈ ℂ. We consider the case where λ lies in the main cardioid of one of the n - 1 principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps Fλ and Fμ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy μ = νj(d+1)λ or where j ∈ ℤ and ν is an (n - 1)th root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.

Sierpinski carpets as Julia sets for imaginary 3-circle inversions

The dynamics of the map is investigated for . This map is derived from multiple geometric circle inversions. Conditions are given for Cantor set and Sierpinski curve Julia sets for this family.

Stylometry and Collaborative Authorship: Eddy, Lovecraft, and 'The Loved Dead'

The authorship of the 1924 short story ‘The Loved Dead’ has been contested by family members of Clifford Martin Eddy, Jr. and Sunand Tryambak Joshi, a leading scholar on Howard Phillips Lovecraft. The authors of this article use stylometric methods to provide evidence for a claim about the authorship of the story and to analyze the nature of Eddy’s collaboration with Lovecraft. Further, we extend Rybicki, Hoover, and Kestemont’s (Collaborative authorship: Conrad, Ford, and rolling delta. Literary and Linguistic Computing , 2014; 29 , 422–31) analysis of stylometry as it relates to collaborations in order to reveal the necessary considerations for employing a stylometric approach to authorial collaboration.