Arithmetic dynamics

Summary

Arithmetic dynamics studies the number-theoretic properties of points under iteration of polynomial and rational maps, and more generally of self-maps of algebraic varieties. Typical questions ask which rational or algebraic points are periodic or preperiodic. Others ask how the height of a point grows along its orbit, and how orbits behave modulo primes or over $p$-adic fields. It combines complex dynamics with Diophantine geometry. Many of its conjectures are dynamical analogues of results on elliptic curves and abelian varieties, where torsion points play the role of preperiodic points.