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Some arithmetic dynamics of diagonally split polynomial maps. (English) Zbl 1391.37073
Summary: Let $$n\geq 2$$, and let $$f$$ be a polynomial of degree at least 2 with coefficients in a number field or a characteristic 0 function field $$K$$. We present two arithmetic applications of a recent theorem of Medvedev-Scanlon to the dynamics of the map $$(f,\ldots,f):(\mathbb P^1_K)^n\to (\mathbb P^1_K)^n$$, 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 points in 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.

##### MSC:
 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 37P15 Dynamical systems over global ground fields 11G35 Varieties over global fields
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