A Galois–Dynamics Correspondence for Unicritical Polynomials

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