An arithmetic approach to parabolic multiplicity in complex dynamics

Abstract: When $ω$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z) = ωz (1 -z)$ and the entire map $F(z) = ωz \mathrm{e}<sup>{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $$F^{\circ n}(z) = z \bigl( 1 +c zⁿ +\mathcal{O}(z^{n+1}) \bigr)$$ 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 $(\mathbb{Z}/p \mathbb{Z})<sup>\times$ is exactly $n$.