Remarks on self-maps with fixed points over a number field

Let $f\colon X\dashrightarrow X$ be a rational self-map with a fixed point $q$, where everything is defined over a number field $K$. We make some remarks on the dynamics of $f$ in a $p$-adic neighbourhood of $q$ for a suitable prime $p$. In particular we show that if the eigenvalues of $Df_q$ are multiplicatively independent, then “most” algebraic points on $X$ have Zariski-dense iterated orbits. (The starting motivation for this was an effort to find an easier proof of the potential density of the variety of lines on a cubic fourfold, due to Voisin and myself. If time permits, I shall also sketch this easier proof.) The talk is based on joint work with Bogomolov and Rovinsky.