A cell decomposition for marked cycle curves

We describe a family ${\text{Cyc}}_p(\mathcal{F})$ 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 ${\text{Per}}_1(0)$ of quadratic polynomials as well as over the family ${\text{Per}}_2(0)$ 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.