Course Outline and Project Descriptions for a Course on Arithmetic Dynamics Arizona Winter School

There are two possibilities for the orbits: • If the orbit Oφ(x) is finite, we say that x is a preperiodic point. • If the orbit Oφ(x) is infinite, we say that x is a wandering point. A important subset of the preperiodic points consists of those points whose orbit eventually return to its starting point. These are called periodic points. Example. Consider the polynomial φ(z) = z − 1 as a map Z → Z. Then 1 −→ 0 −→ ←− −1, so 1 is preperiodic, while 0 and −1 are periodic. It is easy to see that every other element of Z is wandering, since limn→∞ ∣φn(z) ∣∣ = ∞, and more generally, the only φ-preperiodic points in Q are {−1, 0, 1}.