Periodic points of quadratic polynomials in Galois extensions

Robin Zhang

Published 2016-10-04Version 1

For quadratic polynomials with coefficients in a field $k$ of characteristic zero, we seek to understand the periodic points under the actions of its iteration and the absolute Galois group $\mathrm{Gal}(\bar{k}/k)$. Supported by previous numerical work, we conjecture that these two actions are highly correlated. When $k = \mathbb{Q}$, we show that this conjecture implies the non-existence of quadratic periodic points of period $5$ and prove that this conjecture is true for almost all rational $c$ using Hilbert's irreducibility theorem.