2x mod 1 Map

Let $x_0$ be a Rational Number in the Closed Interval $[0,1]$, and generate a Sequence using the Map

$$x_{n+1}\equiv 2x_n{\rm\ (mod\ 1)}.$$ (1)

Then the number of periodic Orbits of period $p$ (for $p$ Prime) is given by

$$N_p={2^p-2\over p}$$ (2)

(i.e, the number of period-$p$ repeating bit strings, modulo shifts). Since a typical Orbit visits each point with equal probability, the Natural Invariant is given by

$$\rho(x)=1.$$ (3)

See also Tent Map

References

Ott, E. Chaos in Dynamical Systems. Cambridge: Cambridge University Press, pp. 26-31, 1993.