Abstract

Classical discrete dynamics is the study of iteration of self-maps of a real or complex object, for example iteration of a morphism $\f:V(\CC)\to V(\CC)$ on the complex points of an algebraic variety. In this setting it turns out that there are natural dynamical analogues to many fundamentals theorems and conjectures in the theory of Diophantine equations and arithmetic geometry. The purpose of this talk is to describe this Diophantine/dynamical analogy and to highlight recent progress and open questions in this comparatively new subject, concentrating for concreteness on the case of a projective morphism $ \f:\PP^{N \to \PP}N $ defined over $\QQ$. Among the topics are:

• Rationality of Periodic Points. To what extent can the periodic or preperiodic points of~$\f$ be $\QQ$-rational? The Diophantine analogue is $\QQ$-rationality of torsion points on elliptic curves (Mazur, Merel) and abelian varieties.

• Integral Points in Orbits. To what extent can the forward $\f$-orbit of a point contain infinitely many integer points? The Diophantine analogue is integral points on curves (Siegel) and on higher dimensional varieties (Faltings, Vojta).

• Dynamical Moduli Spaces. To what extent can the set of morphisms $\PP ^{N \to \PP}N$ modulo “dynamical isomorphism” be given a (nice) algebraic structure. The Diophantine analogues are elliptic modular curves and moduli spaces for abelian varieties.