A cell decomposition for marked cycle curves

TL;DR

This paper introduces a cell decomposition framework for marked cycle curves in dynamical systems, providing algorithms and formulas for quadratic polynomials and rational maps, enhancing understanding of their geometric structure.

Contribution

It develops a canonical cell decomposition for marked cycle curves in specific families of dynamical systems, with explicit algorithms and formulas for their combinatorial properties.

Findings

Algorithms for computing cell decompositions over quadratic polynomial families.

Formulas for the number of cells and genus of the decompositions.

Application to understanding the geometric structure of dynamical cycle spaces.

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.