A Galois-dynamics correspondence for unicritical polynomials

TL;DR

This paper establishes a Galois-dynamics correspondence for unicritical polynomials, linking Galois group actions to polynomial dynamics, and applies it to prove the non-existence of certain quadratic periodic points.

Contribution

It introduces a novel Galois-dynamics correspondence for unicritical polynomials and uses it to derive new results on quadratic periodic points.

Findings

Non-existence of quadratic periodic points of period 5 and 6 for certain quadratic polynomials.

Verification of the correspondence in cases with reducible dynatomic polynomials.

Connection between Galois properties and dynamical behavior of unicritical polynomials.

Abstract

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.