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A cell decomposition for marked cycle curves

Caroline Davis, Malavika Mukundan, Danny Stoll, Giulio Tiozzo

Abstract

We describe a family $\textrm{Cyc}_p(\mathcal{F})$ of marked cycle curves that parameterize the cycles of period $p$ of a given family $\mathcal{F}$ of dynamical systems. We produce algorithms to compute a canonical cell decomposition for the marked cycle curves over the family $\textrm{Per}_1(0)$ of quadratic polynomials as well as over the family $\textrm{Per}_2(0)$ of quadratic rational maps with a critical 2-cycle. We obtain formulas for the number of $d$-cells in these decompositions, giving rise to e.g. a formula for their genus.

A cell decomposition for marked cycle curves

Abstract

We describe a family of marked cycle curves that parameterize the cycles of period of a given family of dynamical systems. We produce algorithms to compute a canonical cell decomposition for the marked cycle curves over the family of quadratic polynomials as well as over the family of quadratic rational maps with a critical 2-cycle. We obtain formulas for the number of -cells in these decompositions, giving rise to e.g. a formula for their genus.

Paper Structure

This paper contains 28 sections, 51 theorems, 128 equations, 13 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Marked cycle curves for $\mathrm{Per}_1(0)$
  3. The cell structure of marked cycle curves for $\mathrm{Per}_1(0)$
  4. Marked cycle curves for $\mathrm{Per}_2(0)$
  5. Examples of rational curves
  6. History
  7. Acknowledgments
  8. Setup
  9. Cycles
  10. Cycle duos
  11. Hyperbolic components and external rays
  12. Kneading data
  13. The cell complex structure of marked cycle curves
  14. The algorithms
  15. The case of $\mathrm{Per}_2(0)$
  16. ...and 13 more sections

Key Result

Theorem 1.2

For each $p \geq 1$ and $m = 1,2$, the Riemann surface $\mathop{\mathrm{Cyc}}\nolimits_p(\mathcal{F}_m)$ has an explicit cell decomposition $\Sigma_{p, m}$ given by gluing sides of a certain union of polygons, with edges labelled by primitive components of period $p$ in the Mandelbrot set, and verti

Figures (13)

  • Figure 1: The cell decomposition for $\mathop{\mathrm{Cyc}}\nolimits_5(\mathcal{F}_1)$, which has genus $2$.
  • Figure 2: The curve $\mathop{\mathrm{Cyc}}\nolimits_3(\mathcal{F}_1)$. The two copies of the Mandelbrot set are joined at the cusp of the airplane component, which is the only primitive component of period $3$.
  • Figure 3: The curve $\mathop{\mathrm{Cyc}}\nolimits_3(\mathcal{F}_1)$ is a blow-up of $\textup{M}_3$ and is the natural domain of the multiplier function, whose contour lines are drawn here. Note that, as expected, the locus where the multiplier has modulus less than $1$ is the union of three hyperbolic components of period $3$.
  • Figure 4: The curve $\mathop{\mathrm{Cyc}}\nolimits_4(\mathcal{F}_1)$, in a rational parameterization.
  • Figure 5: The angles in the proof of \ref{['L:knead']}.
  • ...and 8 more figures

Theorems & Definitions (120)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 110 more