Brunner's theorem

Brunner's theorem relates to the analysis of certain types of dynamical systems, particularly those involving rotations or shifts on a circle. It states that, for a system with a rotational component, the behavior over time can be precisely described by examining the properties of the rotation angle. Specifically, the theorem helps determine whether the system's orbits are evenly spread out (distributed uniformly) or tend to cluster in certain areas, depending on whether the rotation angle is rational or irrational. In essence, Brunner's theorem provides a way to predict the long-term behavior of these systems based on the nature of their rotating motion.