The rotation number is the most robust invariant in circle dynamics. For a deterministic orientation-preserving circle homeomorphism, the correspondence is clean: if the rotation number is an integer, the map has a fixed point. If rational, a periodic orbit. The rotation number and the orbit structure are two descriptions of the same object. You can read one from the other.
Li and Lloyd show that randomness severs this correspondence.
In a random dynamical system on the circle — where the homeomorphism applied at each step is drawn from some distribution — the natural generalization of the rotation number still exists. And random periodic cycles with period two or greater still force the rotation number to be rational. But the fixed-point correspondence breaks. The fiber maps can share a common fixed point while the rotation number is not an integer.
This is not noise blurring a deterministic signal. It is a genuinely different mathematical structure. In the deterministic case, a fixed point pins the dynamics to zero net rotation — the map moves every point and returns it, averaging to zero displacement. In the random case, the fixed point can exist simultaneously with average displacement that is non-integer. The random composition of maps around the fixed point can produce systematic drift that the individual maps, each of which fixes the same point, do not.
The converse partially survives: if the mean rotation number equals an integer, the fiber maps possess a fixed point with positive probability. Not certainty — probability. The deterministic guarantee degrades to a probabilistic statement. The rigid connection between topology (fixed points) and dynamics (rotation) loosens into a statistical one.
This matters because rotation numbers are used across dynamical systems theory as diagnostic tools. They classify quasiperiodic motion, detect mode-locking, characterize phase oscillators. The assumption that they faithfully report the orbit structure underlies their use. In random systems, they report the average dynamics but not the pointwise structure. The number is the same kind of object; what it certifies is weaker.
Noise does not perturb the invariant. It changes what the invariant means.