Some Arithmetic Dynamics of Diagonally Split Polynomial Maps

Abstract

Let $K$ be a number field or the function field of a curve over an algebraically closed field of characteristic 0. Let $n\geq 2$, and let $f(X)\in K[X]$ be a polynomial of degree $d\geq 2$. We present twoarithmetic properties of the dynamics of the coordinate-wise self-map $\varphi=f\times\ldots\times f$ of $(\bP^1)^n$, namely the dynamical analogs of the Hasse principle and the Bombieri-Masser-Zannier height bound theorem. In particular, we prove that the Hasse principle holds when we intersect an orbit and a preperiodic subvariety, and that the intersection of a curve with the union of all periodic hypersurfaces have bounded heights unless that curve is vertical or contained in a periodic hypersurface. A common crucial ingredient for the proof of these two properties is a recent classification of $\varphi$-periodic subvarieties by Medvedev-Scanlon. We also present the problem of primitive prime divisors in dynamical sequences by Ingram-Silverman which is needed and closely related to the dynamical Hasse principle. Further questions on the bounded height result, and a possible generalization of the Medvedev-Scanlon classification are briefly given at the end.