A solution to the degree-d twisted rabbit problem

TL;DR

This paper generalizes the twisted rabbit problem from quadratic to degree d polynomials, providing a solution based on the d^2-adic expansion of the twisting element, extending previous quadratic results.

Contribution

It introduces a new solution method for the degree-d twisted rabbit problem, expanding the understanding of polynomial twisting in complex dynamics.

Findings

Solution depends on d^2-adic expansion of the twisting element

Generalizes previous quadratic twisted rabbit problem results

Provides explicit classification for degree d polynomials

Abstract

We solve generalizations of Hubbard's twisted rabbit problem for analogues of the rabbit polynomial of degree $d\geq 2$. The twisted rabbit problem asks: when a certain quadratic polynomial, called the Douady Rabbit polynomial, is twisted by a cyclic subgroup of a mapping class group, to which polynomial is the resulting map equivalent (as a function of the power of the generator)? The solution to the original quadratic twisted rabbit problem, given by Bartholdi--Nekrashevych, depended on the 4-adic expansion of the power of the mapping class by which we twist. In this paper, we provide a solution that depends on the $d^2$-adic expansion of the power of the mapping class element by which we twist.