Dyadic transformation

A dyadic transformation is a mapping <math>d: \R \to \R^\infty, <math> produced by the rule <math>x_0 \in [0, 1]<math> and <math>x_{i+1} = 2x_i \mod 2<math> for all i.

If <math>x_0<math> is rational, the image will be a finite subset of <math>[0, 1]<math> (for odd denominators a finite cyclic group), for irrational ones a ?dense subset?.