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Transitivity of the pure Hurwitz classes of quadratic post-critically finite polynomials

Yvon Verberne, Rebecca R. Winarski

TL;DR

The paper proves a transitivity property for pure Hurwitz classes of quadratic PCF polynomials by constructing an explicit finite sequence of portrait-modifying moves, realized as mapping class actions, that transform one map into another up to pure Hurwitz equivalence. Central to the method is an algorithm that navigates the space of dynamical portraits using a repertoire of half-twist induced functions, increasing or decreasing cycle and pre-period lengths, and reconfiguring the critical-point structure while preserving quadratic compatibility. The termination of the algorithm guarantees a finite marked set and a mapping class linking the two portraits, thus establishing the main result. This approach connects holomorphic dynamics, geometric group theory, and surface topology, and opens paths toward extending to higher degree Hurwitz classes and impure mapping class actions.

Abstract

We prove that for two post-critically finite quadratic polynomials $f,g$, there is a mapping class $φ$ of the sphere with finitely many marked points such that $fφ$ and $g$ are pure Hurwitz equivalent.

Transitivity of the pure Hurwitz classes of quadratic post-critically finite polynomials

TL;DR

The paper proves a transitivity property for pure Hurwitz classes of quadratic PCF polynomials by constructing an explicit finite sequence of portrait-modifying moves, realized as mapping class actions, that transform one map into another up to pure Hurwitz equivalence. Central to the method is an algorithm that navigates the space of dynamical portraits using a repertoire of half-twist induced functions, increasing or decreasing cycle and pre-period lengths, and reconfiguring the critical-point structure while preserving quadratic compatibility. The termination of the algorithm guarantees a finite marked set and a mapping class linking the two portraits, thus establishing the main result. This approach connects holomorphic dynamics, geometric group theory, and surface topology, and opens paths toward extending to higher degree Hurwitz classes and impure mapping class actions.

Abstract

We prove that for two post-critically finite quadratic polynomials , there is a mapping class of the sphere with finitely many marked points such that and are pure Hurwitz equivalent.

Paper Structure

This paper contains 10 sections, 5 theorems, 9 figures.

Table of Contents

  1. Introduction
  2. Future directions
  3. Outline of paper
  4. Acknowledgments
  5. Portrait features
  6. Abstract portraits
  7. Portrait features
  8. Portrait functions
  9. Algorithm
  10. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $f,g:S^2\rightarrow S^2$ be marked quadratic branched covers with dynamical portraits $\Gamma_f$ and $\Gamma_g$. There exists a finite set of points $M\subset S^2$ that contains the post-critical set for $f$ and $g$ and $\phi\in\mathop{\mathrm{Mod}}\nolimits(S^2,M)$ so that $f\circ \phi$ is pure

Figures (9)

  • Figure 1: When a portrait has two disjoint pre-periods, we measure the length of each pre-period $r_1-1$ and $r_2-1$ and the distances between the pre-periods, $k_1$ and $k_2$.
  • Figure 2: The function \ref{['Function:CriticalPointsInCycle']}(L) increases the position of $C_2$ in the orbit of $C_1$ and changes no other portrait features.
  • Figure 3: Applying Function \ref{['Function:CriticalPointFromCycleTopre-period']} creates a new function where $f^{r-1}(C_1)$ maps to $C_2$.
  • Figure 4: Applying Function \ref{['Function:CriticalPointpre-periodToCycle']} to the critical point $C_j$ creates a new function in for which $C_j$ periodic.
  • Figure 5: Function \ref{['Function:Merge']} may change all portrait features.
  • ...and 4 more figures

Theorems & Definitions (22)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 12 more