We study the behavior of canonical height functions , associated to rational maps f, on totally p-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of on the maximal totally p-adic field if the map f has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset X in the compositum of all number fields of degree at most d such that f(X) = X for some non-linear polynomial f. This answers a question of W. Narkiewicz from 1963.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-3-5,
author = {Lukas Pottmeyer},
title = {Heights and totally p-adic numbers},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {277-291},
zbl = {06498811},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-3-5}
}
Lukas Pottmeyer. Heights and totally p-adic numbers. Acta Arithmetica, Tome 168 (2015) pp. 277-291. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-3-5/