Invariant varieties for polynomial dynamical systems

Abstract

We study algebraic dynamical systems (and, more generally, s -varieties) F:A n C ?A n 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 s:C?C those algebraic varieties X?A n C for which F(X)?X s . As a special case, we show that if f(x)?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?A 2 C is an irreducible curve that is invariant under the action of (x,y)?(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 s(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 s , 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 s