Abstract: In dynamics we study orbits of points under iteration of a map f : X --> X. In number theory we study rational and algebraic numbers, and height functions that measure the complexity these numbers play an important role. In this talk I will discuss the interaction of height functions and iteration of rational maps on varieties, including: (1) The theory of canonical heights, which interact especially nicely with (polarized) morphisms of projective varieties. (2) A coarser measure of complexity called the arithmetic degree, which exists for more general maps and is related to the geometric complexity (dynamical degree) of the map f.
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