# Invariant varieties for polynomial dynamical systems - Mathematics > Dynamical Systems

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Abstract: We study algebraic dynamical systems and, more generally,$\sigma$-varieties $\Phi:{\mathbb A}^n {\mathbb C} \to {\mathbb A}^n {\mathbbC}$ given by coordinatewise univariate polynomials by refining a theorem ofRitt. More precisely, we find a nearly canonical way to write a polynomial as acomposition of -clusters-. Our main result is an explicit description of theweakly skew-invariant varieties. As a special case, we show that if $fx \in{\mathbb C}x$ is a polynomial of degree at least two which is not conjugateto a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and $X\subseteq {\mathbb A}^2 {\mathbb C}$ is an irreducible curve which is invariantunder the action of $x,y \mapsto fx,fy$ and projects dominantly in bothdirections, then $X$ must be the graph of a polynomial which commutes with $f$under composition. As consequences, we deduce a variant of a conjecture ofZhang on the existence of rational points with Zariski dense forward orbits anda strong form of the dynamical Manin-Mumford conjecture for liftings of theFrobenius.We also show that in models of ACFA$ 0$, a disintegrated set defined by$\sigmax = fx$ for a polynomial $f$ has Morley rank one and is usuallystrongly minimal, that model theoretic algebraic closure is a locally finiteclosure operator on the nonalgebraic points of this set unless theskew-conjugacy class of $f$ is defined over a fixed field of a power of$\sigma$, and that nonorthogonality between two such sets is definable infamilies if the skew-conjugacy class of $f$ is defined over a fixed field of apower of $\sigma$.

Author: ** Alice Medvedev, Thomas Scanlon**

Source: https://arxiv.org/