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, Tm(Tn(x))=Tm n(x), since Tn(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.
Read more about this topic: Iterated Function
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