Invariant varieties for polynomial dynamical systems

Abstract

We study algebraic dynamical systems (and, more generally, $\sigma$-varieties) $\Phi:\mathbb{A}^n_\mathbb{C} \to \mathbb{A}^n_\mathbb{C}$ given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of “clusters” from which one may easily read off possible compositional identities. Our main result is an explicit description of the (weakly) skew-invariant varieties, that is, for a fixed field automorphism $\sigma:\mathbb{C} \to \mathbb{C}$ those algebraic varieties $X \subseteq \mathbb{A}^n_\mathbb{C}$ for which $\Phi(X) \subseteq X^\sigma$. As a special case, we show that if $f(x) \in \mathbb{C}[x]$ is a polynomial of degree at least two that is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and $X \subseteq \mathbb{A}^2_\mathbb{C}$ is an irreducible curve that is invariant under the action of $(x,y) \mapsto (f(x),f(y))$ and projects dominantly in both directions, then $X$ must be the graph of a polynomial that commutes with $f$ under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius.
We also show that in models of ACFA$_0$, a disintegrated set defined by $\sigma(x) = f(x)$ for a polynomial $f$ has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew-conjugacy class of $f$ is defined over a fixed field of a power of $\sigma$, and that nonorthogonality between two such sets is definable in families if the skew-conjugacy class of $f$ is defined over a fixed field of a power of $\sigma$.