Polynomial dynamics

Details Polynomial dynamics Polynomial dynamics

2009-01-15

arxiv.org/abs/0901.2352v2

Mathematics

Dynamical Systems

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Scientific paper

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, $\Phi(x_1,...,x_n) = (f_1(x_1),...,f_n(x_n))$ by refining an old theorem of Ritt on compositional identities amongst polynomials. Our main result is an explicit description of the skew-invariant varieties, that is, for a field automorphism $\sigma:{\mathbb C} \to {\mathbb C}$ we describe those algebraic varieties $X \subseteq {\mathbb A}^n_{\mathbb C}$ such that $\Phi(X) = X^\sigma$. In particular, taking $\sigma$ to be the identity function, we characterize the $\Phi$-invariant varieties. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits, a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius, and an answer to the question of definability of nonorthogonality for minimal types in difference closed fields of characteristic zero extending the formula $\sigma(x) = f(x)$ for $f$ a polynomial.

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