Integrality properties of Böttcher coordinates for one-dimensional superattracting germs

Abstract

Let $R$ be a ring of characteristic $0$ with field of fractions $K$, and let $m\ge 2$. The Böttcher coordinate of a power series $\phi (x)\in x^m+x^{m+1}R[[x]]$ is the unique power series $f_{\phi }(x)\in x+x^{2}K[[x]]$ satisfying $\phi \circ f_{\phi }(x)=f_{\phi }(x^m)$. In this paper we study the integrality properties of the coefficients of $f_{\phi }(x)$, partly for their intrinsic interest and partly for potential applications to $p$-adic dynamics. Results include: (1) If $p$ is prime and $R={\mathbb{Z}}_p$ and $\phi (x)\in x^p+p{x}^{p+1}R[[x]]$, then $f_{\phi }(x)\in R[[x]]$. (2) If $\phi (x)\in x^m+m{x}^{m+1}R[[x]]$, then $f_{\phi }(x)=x{\sum }_{k=0}^{\infty }{a}_k{x}^k/k!$ with all $a_k\in R$. (3) In (2), if $m=p^2$, then $a_k\equiv -1(\mathrm{mod}p)$ for all $k$ that are powers of $p$.

arXiv comment27 pages. Version 2 fixes the statement and proof of Theorem 4