关于Pappus-Steiner映射的动力学
On the dynamics of the Pappus-Steiner map
摘要：
我们从经典射影几何中的Pappus和Steiner定理中提取出一个二维动力系统。我们计算了这个系统的显式公式，并研究了它的基本几何性质。然后，我们使用阿丁互易性来证明所有足够大的素数$p$，该系统承认周期点的订单$3$和$4$在字段${\mathbb F}_p$，这导致了一个意想不到的伽罗瓦理论猜想$n$ -周期点。我们也给出了一个简短的讨论Leisenring定理，并表明，它导致相同的动力系统的Pappus-Steiner定理。附录中包含了计算机辅助分析这个系统在领域的实数。
英文摘要：
We extract a two-dimensional dynamical system from the theorems of Pappus and Steiner in classical projective geometry. We calculate an explicit formula for this system, and study its elementary geometric properties. Then we use Artin reciprocity to characterise all sufficiently large primes $p$ for which this system admits periodic points of orders $3$ and $4$ over the field ${\mathbb F}_p$; this leads to an unexpected Galois-theoretic conjecture for $n$-periodic points. We also give a short discussion of Leisenring's theorem, and show that it leads to the same dynamical system as the Pappus-Steiner theorem. The appendix contains a computer-aided analysis of this system over the field of real numbers.
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