  The dynatomic modular curves parametrize polynomial maps together with a
point of period $n$. It is known that the dynatomic curves $Y_1(n)$ are smooth
and irreducible in characteristic 0 for families of polynomial maps of the form
$f_c(z) = z^m +c$ where $m\geq 2$. In the present paper, we build on the work
of Morton to partially characterize the primes $p$ for which the reduction
modulo $p$ of $Y_1(n)$ remains smooth and/or irreducible. As an application, we
give new examples of good reduction of $Y_1(n)$ for several primes dividing the
ramification discriminant when $n=7,8,11$. The proofs involve arithmetic and
complex dynamics, reduction theory for curves, ramification theory, and the
combinatorics of the Mandelbrot set.
