Rotation map defines dynamical systems on the circle ( maps of the circle to itself = self-maps on the circle). The theory describing it is called Rotation Theory^[1]

     "If a is rational, then every point is periodic. If a is irrational, then every point has a dense orbit." David Richeson^[2]

rational

Rotation map $R$ describes counterclockwise rotation of point $\theta$ thru ${\frac {p}{q}}$ turns on the unit circle:

  $R_{\frac {p}{q}}(\theta )=\theta +{\frac {p}{q}}$

It is used for computing:

itinerary

rotation maps

irrational

References

[1] An Introduction to Rotation Theory by Christian Kuehn

[2] three-cool-facts-about-rotations-of-the-circle by David Richeson

[1]

[2]