An arithmetic approach to parabolic multiplicity in complex dynamics

Abstract

When $\omega$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z)=\omega z(1-z)$ and the entire map $F(z)=\omega z{\mathrm{e}}^{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $F^{\circ n}(z)=z(1+c{z}^n+\mathcal{O}(z^{n+1}))$ with $c\ne 0$. 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 $n$.

26 pages