Integrality properties of Böttcher coordinates for one-dimensional superattracting germs

Let $R$ be a ring of characteristic $0$ with field of fractions $K$ and let $m\geq 2$ . The Böttcher coordinate of a power series $\unicode[STIX]{x1D711}(x)\in x^{m}+x^{m+1}R\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$ is the unique power series $f_{\unicode[STIX]{x1D711}}(x)\in x+x^{2}K\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$ satisfying $\unicode[STIX]{x1D711}\circ f_{\unicode[STIX]{x1D711}}(x)=f_{\unicode[STIX]{x1D711}}(x^{m})$ . In this paper we study the integrality properties of the coefficients of $f_{\unicode[STIX]{x1D711}}(x)$ , 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 $\unicode[STIX]{x1D711}(x)\in x^{p}+px^{p+1}R\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$ , then $f_{\unicode[STIX]{x1D711}}(x)\in R\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$ . (2) If $\unicode[STIX]{x1D711}(x)\in x^{m}+mx^{m+1}R\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$ , then $f_{\unicode[STIX]{x1D711}}(x)=x\sum _{k=0}^{\infty }a_{k}x^{k}/k!$ with all $a_{k}\in R$ . (3) In (2), if $m=p^{2}$ , then $a_{k}\equiv -1~\text{(mod}~p\text{)}$ for all $k$ that are powers of $p$ .