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

TL;DR

This paper explores the complex dynamics of generalized McMullen maps, revealing new behaviors in their Julia sets, including intricate structures formed by quadratic Julia sets and combinatorial modifications.

Contribution

It introduces a combinatorial model for the dynamics of generalized McMullen maps with bounded critical orbits, highlighting new Julia set structures.

Findings

Julia sets contain infinitely many quadratic Julia sets

Existence of Julia sets with modified external ray identifications

New dynamical behaviors in generalized McMullen maps

Abstract

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.