Arithmetic Dynamics - Definitions and Notation From Discrete Dynamics

Definitions and Notation From Discrete Dynamics

Let S be a set and let F : S → S be a map from S to itself. The iterate of F with itself n times is denoted

F^{(n)} = F \circ F \circ \cdots \circ F.

A point P ∈ S is periodic if F(n)(P) = P for some n > 1.

The point is preperiodic if F(k)(P) is periodic for some k ≥ 1.

The (forward) orbit of P is the set

O_F(P) = \bigl\{ P, F(P), F^{(2)}(P), F^{(3)}(P), F^{(4)}(P), \ldots\bigr\}.

Thus P is preperiodic if and only if its orbit OF(P) is finite.

Read more about this topic:  Arithmetic Dynamics

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