Dyadic transformation
The dyadic transformation (also known as the dyadic map, bit shift map, 2x mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) T : [ 0 , 1 ) → [ 0 , 1 ) ∞ {\displaystyle T:[0,1)\to [0,1)^{\infty }} x ↦ ( x 0 , x 1 , x 2 , … ) {\displaystyle x\mapsto (x_{0},x_{1},x_{2},\ldots )} (where [ 0 , 1 ) ∞ {\displaystyle [0,1)^{\infty }} is the set of sequences from [ 0 , 1 ) {\displaystyle [0,1)} ) produced by the rule x 0 = x {\displaystyle x_{0}=x} for all n ≥ 0 , x n + 1 = ( 2 x n ) mod 1 {\displaystyle {\text{for all }}n\geq 0,\ x_{n+1}=(2x_{n}){\bmod {1}}} . Equivalently, the dyadic transformation can also be defined as the iterated function map of the piecewise linear function T ( x ) = { 2 x 0 ≤ x < 1 2 2 x − 1 1 2 ≤ x < 1.