多项式动力系统的不变簇
Invariant varieties for polynomial dynamical systems
摘要：
我们研究代数动力系统（更一般地说，$σ$ -品种）$Φ:{\mathbb A}^n_{\mathbb C} \to {\mathbb A}^n_{\mathbb C}$由coordinatewise单变量多项式精炼的Ritt定理。更确切地说，我们找到了一个几乎规范的方式来写一个多项式作为一个组成的“集群”。我们的主要结果是一个明确的描述（弱）斜不变品种。作为一种特殊情况，我们证明了：如果$f(x) \in {\mathbb C}[x]$是一个不共轭于单式、Chebyshev多项式或负Chebyshev多项式的至少二次多项式，并且$X \subseteq {\mathbb A}^2_{\mathbb C}$是一条不可约曲线，在$(x,y) \mapsto (f(x),f(y))$的作用下是不变的，并且在两个方向上都有优势投影，那么$X$一定是一个与$f$可交换的多项式的合成图.作为结果，我们推导出一个变种的猜想张的存在合理的点与Zebraki密集的前向轨道和一个强大的形式的动力学Manin-Mumford猜想提升的Frobenius。我们还证明了在ACFA $_0$模型中，由$σ(x) = f(x)$定义的多项式$f$的分解集具有Morley秩1，并且通常是强极小的，除非$f$的斜共轭类定义在$σ$的幂的固定域上，否则模型理论代数闭包是该集合的非代数点上的局部有限闭包算子，如果$f$的斜共轭类定义在$σ$的幂的固定域上，则两个这样的集合之间的非正交性在族中是可定义的。
英文摘要：
We study algebraic dynamical systems (and, more generally, $σ$-varieties) $Φ:{\mathbb A}^n_{\mathbb C} \to {\mathbb A}^n_{\mathbb C}$ 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 $σ(x) = f(x)$ 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 $σ$, 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 $σ$.
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