Good and bad reduction of dynatomic modular curves

The dynatomic modular curves parameterize one-parameter families of dynamical systems on ${\bf P}^1$ along with periodic points (or orbits). These are analogous to the standard modular curves parameterizing elliptic curves with torsion points (or subgroups). For the family $x^2 + c$ of quadratic dynamical systems, the corresponding modular curves are smooth in characteristic zero. We give several results about when these curves have good/bad reduction to characteristic p, as well as when the reduction is irreducible. These results are motivated by uniform boundedness conjectures in arithmetic dynamics, which will be explained.