Chapter 4: Systems Based on Numbers

Section 7: Iterated Maps and the Chaos Phenomenon

Exact iterates [in iterated maps]

For any integer a the nth iterate of x FractionalPart[a x] can be written as FractionalPart[an x], or equivalently 1/2 - ArcTan[Cot[an π x]]/π. In the specific case a = 2 the iterates of If[x < 1/2, a x, a (1 - x)] have the form ArcCos[Cos[2n π x]]/π. (See pages 903 and 1098.)

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]