Iterated Function - Abelian Property and Iteration Sequences

Abelian Property and Iteration Sequences

In general, the following identity holds for all non-negative integers m and n,

This is structurally identical to the property of exponentiation that aman = am+n, i.e. the special case f(x)=ax.

In general, for arbitrary general (negative, non-integer, etc.) indices m and n, this relation is called the translation functional equation, cf. Schröder's equation. On a logarithmic scale, this reduces to the nesting property of Chebyshev polynomials, T_m(T_n(x))=T_{m n}(x), since T_n(x) = cos(n arcos(x )).

The relation (f m )n(x) = (f n )m(x) = f mn(x) also holds, analogous to the property of exponentiation that (am )n = (an )m = amn.

The sequence of functions f n is called a Picard sequence, named after Charles Émile Picard.

For a given x in X, the sequence of values f n(x) is called the orbit of x.

If f n (x) = f n+m (x) for some integer m, the orbit is called a periodic orbit. The smallest such value of m for a given x is called the period of the orbit. The point x itself is called a periodic point. The cycle detection problem in computer science is the algorithmic problem of finding the first periodic point in an orbit, and the period of the orbit.