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A solution to the degree-d twisted rabbit problem

Malavika Mukundan, Rebecca R. Winarski

TL;DR

This paper provides a solution to Hubbard's twisted rabbit problem that depends on the $d^2$-adic expansion of the power of the mapping class element by which the authors twist.

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.

A solution to the degree-d twisted rabbit problem

TL;DR

This paper provides a solution to Hubbard's twisted rabbit problem that depends on the -adic expansion of the power of the mapping class element by which the authors twist.

Abstract

We solve generalizations of Hubbard's twisted rabbit problem for analogues of the rabbit polynomial of degree . 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 -adic expansion of the power of the mapping class element by which we twist.
Paper Structure (7 sections, 6 theorems, 14 equations, 5 figures, 1 table)

This paper contains 7 sections, 6 theorems, 14 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Hubbard trees and degree-d polynomials
  3. Hubbard trees
  4. The degree-d polynomials with a 3-periodic critical cycle
  5. Reduction Formulae
  6. Invariant trees
  7. Proof of Main Theorem

Key Result

Theorem 1.1

Let $R_d$ be the degree-$d$ rabbit polynomial and $D_x$ the Dehn twist about the curve $x=x_d$. If $(d+1)| m_i$ for all $i \in \{1,2,...,\sigma_{d^2}(m)\}$, then Otherwise, let $i$ be the least index such that $m_i$ is not divisible by $d+1$. We may write $m_i$ uniquely as $d\ell+n$, where $\ell,n \in \{0,1,...,d-1\}$. Since $(d+1)\nmid m_i$, we have that $\ell\neq n$. Then:

Figures (5)

  • Figure 1: The Hubbard tree of the rabbit polynomial along with the simple closed curves $x$, $y$ and $z$
  • Figure 2: The Julia sets and Hubbard trees for the unicritical polynomials of degree $5$ with 3-periodic critical point.
  • Figure 4: Left: The Hubbard tree $H_-$ for $D^i_yR_d$ with $-(d-1)\leq i\leq -1$. Center: a tree that is homotopic to $D_y(H_-)$. Right: a tree that is homotopic to the preimage of $D_y(H_-)$ under $R_5$.
  • Figure 5: Left: The Hubbard tree $H_+$ for $D^i_yR_d$ with $1\leq i\leq d-1$. Center: a tree that is homotopic to $D_y^{-1}(H_+)$. Right: a tree that is homotopic to the preimage of $D_y^{-1}(H_+)$ under $R_5$.
  • Figure 6: Left: The Hubbard tree $H$ for $D_x^{-1}R_d$, Center: $D_x(H)$, Right: $R_5^{-1}(D_x(H))$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 1 more