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Integrality properties of Böttcher coordinates for one-dimensional superattracting germs

Published online by Cambridge University Press:  06 July 2018

ADRIANA SALERNO  and
JOSEPH H. SILVERMAN
ADRIANA SALERNO
Affiliation:
Department of Mathematics, Bates College, Lewiston, ME 04240, USA email [email protected]
JOSEPH H. SILVERMAN
Affiliation:
Mathematics Department, Box 1917, Brown University, Providence, RI 02912, USA email [email protected]
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Abstract

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 $p$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 1 , January 2020 , pp. 248 - 271
Copyright
© Cambridge University Press, 2018 

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