Exact iterates [in iterated maps]

For any integer a the n^{th} iterate of x FractionalPart[a x] can be written as FractionalPart[a^{n} x], or equivalently 1/2 - ArcTan[Cot[a^{n} 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[2^{n} π x]]/π. (See pages 903 and 1098.)