论文标题
代数群体的内态动力学
Dynamics of endomorphisms of algebraic groups
论文作者
论文摘要
令$σ$表示有限场的代数关闭上平滑代数$ g $的内态性，并假设所有迭代$σ$都有许多固定点有限的。 Steinberg给出了$σ$的固定点数量（因此，其所有迭代的$σ^n $）在半神经案例中提供了公式，从而代表了其Artin-Mazur Zeta函数作为理性函数的表示。我们将其推广到任意（平滑）代数组$ g $，其中固定点的数量$σ_n$的$σ^n $可能取决于$ n $的$ p $ - adiC属性。我们通过“有限的变形”（FAD-）序列的概念来使序列$（σ_n）$的结构进行公理。这种序列也发生在拓扑动力学中，我们关于Zeta功能和轨道渐近计数的后续结果在这种情况下同样适用。例如，到$ s $ integer动态系统，添加剂蜂窝自动机和其他紧凑型阿伯利亚组。我们证明了相关的Artin-Mazur Zeta函数的二分法，并研究了质数定理的类似物，以计算长度$ \ leq n $的函数定期轨道。对于一个代数组$ g $，我们通过$ g $的$ \ ell $ - adiC同胞zeta函数表示错误项。
Let $σ$ denote an endomorphism of a smooth algebraic group $G$ over the algebraic closure of a finite field, and assume all iterates of $σ$ have finitely many fixed points. Steinberg gave a formula for the number of fixed points of $σ$ (and hence of all of its iterates $σ^n$) in the semisimple case, leading to a representation of its Artin-Mazur zeta function as a rational function. We generalise this to an arbitrary (smooth) algebraic group $G$, where the number of fixed points $σ_n$ of $σ^n$ can depend on $p$-adic properties of $n$. We axiomatise the structure of the sequence $(σ_n)$ via the concept of a `finite-adelically distorted' (FAD-)sequence. Such sequences also occur in topological dynamics, and our subsequent results about zeta functions and asymptotic counting of orbits apply equally well in that situation; for example, to $S$-integer dynamical systems, additive cellular automata and other compact abelian groups. We prove dichotomies for the associated Artin-Mazur zeta function, and study the analogue of the Prime Number Theorem for the function counting periodic orbits of length $\leq N$. For an algebraic group $G$ we express the error term via the $\ell$-adic cohomological zeta function of $G$.
