arXiv subjects

Kelsey Brouwer

Publications and source records attributed to Kelsey Brouwer.

3 recordsLinked to original sources

arXiv2026

Paper Fortune Tellers in Julia sets of Generalized McMullen maps II: Sidecars and Zippers

We study the family of complex rational functions known as Generalized McMullen maps, F(z) = z^n + a/z^n+b, for integer n at least 3 fixed, and complex parameters a, b with a nonzero. In prior work by the same authors, we provided a combinatorial model for a large class of maps whose Julia sets contain both infinitely many homeomorphic copies of quadratic Julia sets conjugate to the ``basilica'', and infinitely many subsets homeomorphic to a set which is obtained by starting with the basilica, then changing a finite number of pairs of external ray landing point identifications, following an algorithm we described. In this article, we generalize beyond the basilica, and provide a catalog of additional types of hyperbolic Julia sets of Generalized McMullen maps, where the ``baby'' Julia set can be any rabbit, aeroplane, or Kokopelli quadratic Julia set; that is, where the c-value can be taken from any bulb attached to the main cardioid of the Mandelbrot set, or from the main cardioid of any principal baby Mandelbrot set (no renormalizations).

math.DS↗

arXiv2026

Exploring baby Julia sets in parameter space slices for Generalized McMullen Maps

For the family of complex rational functions of the form R(z)= z^n + a/z^n+b, known as "Generalized McMullen maps", for non-zero a, and integer n fixed and at least 3, we describe the apparent phenomena of baby Julia sets in parameter space appearing both in slices with independent critical orbits and a slice defined by imposing a critical orbit relation. Specifically, we introduce the subfamily where one of two critical orbits is set to be a super-attracting fixed point, provide some general results on this subfamily and describe how Julia set copies in the parameter space slice occur--due to parameters for which the other critical orbit is in the (not immediate) basin of attraction of this fixed critical point. We provide several conjectures on this intriguing phenomena to catalyze further study.

math.DS↗

arXiv2025

Paper Fortune Tellers in the combinatorial dynamics of some generalized McMullen maps with both critical orbits bounded

For the family of complex rational functions known as "Generalized McMullen maps", F(z) = z^n + a/z^n+b, for complex parameters a and b, with a nonzero, and any integer n at least 3 fixed, we reveal, and provide a combinatorial model for, some new dynamical behavior. In particular, we describe a large class of maps whose Julia sets contain both infinitely many homeomorphic copies of quadratic Julia sets and infinitely many subsets homeomorphic to a set which is obtained by starting with a quadratic Julia set, then changing a finite number of pairs of external ray landing point identifications, following an algorithm we will describe.

math.DS↗