  In an analogy with the Galois homothety property for torsion points of
abelian varieties that was used in the proof of the Mordell-Lang conjecture, we
describe a correspondence between the action of a Galois group and the
dynamical action of a rational map. For nonlinear polynomials with rational
coefficients, the irreducibility of the associated dynatomic polynomial serves
as a convenient criterion, although we also verify that the correspondence
occurs in several cases when the dynatomic polynomial is reducible. The work of
Morton, Morton-Patel, and Vivaldi-Hatjispyros in the early 1990s connected the
irreducibility and Galois-theoretic properties of dynatomic polynomials to
rational periodic points; from the Galois-dynamics correspondence, we derive
similar consequences for quadratic periodic points of unicritical polynomials.
This is sufficient to deduce the non-existence of quadratic periodic points of
quadratic polynomials with exact period 5 and 6, outside of a specified finite
set from Morton and Krumm's work in explicit Hilbert irreducibility.
