 Cohomological equation

For a map $T:X\to X$ and a given dynamical observable $f$, the additive cohomological equation asks for a function $u$ satisfyingWhen such $u$ exists, $f$ is a difference accumulated between consecutive states. Summing along a trajectory makes the intermediate terms cancel:Thus bounded $u$ forces these sums to remain bounded, and any periodic orbit of period $n$ must have sum zero. These are necessary conditions, not a universal existence theorem.

For the circle rotation $T(x)=x+\alpha (\mathrm{mod}1)$, try $f(x)=e^{2\pi ikx}$ and $u(x)=c{e}^{2\pi ikx}$. Then $c=(e^{2\pi ik\alpha }-1{)}^{-1}$ when the denominator is nonzero. If it vanishes, this mode is an obstruction; if it is very small, the coefficient is large. This explains how a simple equation leads to delicate regularity questions in dynamical systems.