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)

produced by the rule

Equivalently, the dyadic transformation can also be defined as the iterated function map of the piecewise linear function

The name bit shift map arises because, if the value of an iterate is written in binary notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero.

The dyadic transformation provides an example of how a simple 1-dimensional map can give rise to chaos.

Relation to tent map and logistic map

The dyadic transformation is topologically conjugate to :

The r = 4 case of the logistic map is ; this is related to the bit shift map in variable x by

There is semi-conjugacy between the dyadic transformation (here named angle doubling map) and the quadratic polynomial. Here map doubles angles measured in turns.