2x mod 1 Map

The mod 1 map, also called the doubling map, generates a sequence from an initial rational number in the closed interval using the map

(1)

Then the number of periodic map orbits of period (for prime) is given by

(2)

(i.e., the number of period- repeating bit strings, modulo shifts). Since a typical map orbit visits each point with equal probability, the natural invariant is given by

(3)