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Invariant Varieties for Polynomial Dynamical Systems by Medvedev, Alice; Scanlon, Thomas is a scholarly article available to read on EtoBox.

What is Invariant Varieties for Polynomial Dynamical Systems about?

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 a theorem of Ritt. More precisely, we find a nearly canonical way to write a polynomial as a composition of "clusters". Our main result is an explicit description of the (weakly) skew-invariant varieties. As a special case, we show that if $f(x) \in {\mathbb C}[x]$ is a polynomial of degree at least two which 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 which 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 which 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 $f$

Author

Medvedev, Alice; Scanlon, Thomas

Published

2009

Language

EN