AI 中文总结
针对复动力学中本原n次单位根对应的两类映射的抛物重数问题，提出一种适用于多项式和超越性情形的算术证明方法，分别在p进域与整数模n-1环中开展工作。
AI 中文摘要
当ω是本原n次单位根时，二次多项式F(z)=ωz(1−z)和整函数F(z)=ωze⁻^z在0处都有一个抛物不动点，它们的抛物重数等于1，即F°ⁿ(z)=z(1+czⁿ+O(zⁿ⁺¹))且c≠0。该事实的经典证明是超越性的，我们提出一种算术证明：在超越性情形下，该证明可从《Towards global models near homoclinic tangencies of dissipative diffeomorphisms》（作者H. Broer、C. Simó、J.C. Tatjer）中提取，且需在Z/(n−1)Z中进行；在多项式情形下，该证明是新的，且需在合适素数p的p进域Qₚ中进行，其中(Z/pZ)^×中2的阶恰好为n。
英文摘要
When $ω$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z) = ωz (1 -z)$ and the entire map $F(z) = ωz \mathrm{e}^{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $F^{\circ n}(z) = z \bigl( 1 +c z^n +\mathcal{O}(z^{n+1}) \bigr)$ with $c \neq 0$. The classical proof of this fact is transcendental. We present an arithmetic proof which may be extracted from [Towards global models near homoclinic tangencies of dissipative diffeomorphisms; H. Broer, C. Simó, J.C. Tatjer] in the transcendental case and requires working in $\mathbb{Z}/(n -1) \mathbb{Z}$, and which is new in the polynomial case and requires working in the $p$-adic field $\mathbb{Q}_p$ for a suitable prime $p$ such that the order of $2$ in $(\mathbb{Z}/p \mathbb{Z})^\times$ is exactly $n$.
Comments26 pages
